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presented  to  the 
UNIVERSITY  LIBRARY 
UNIVERSITY  OF  CALIFORNIA 
SAN  DIEGO 

by 


Dr.  George  McEwen 


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THEORY  AND   PEACTICE 


OF 


INTERPOLATION: 


INCLUDING 


Mechanical   Quadrature,    and    Other   Important    Problems 
Concerned  with  the  Tabular  Values  of  Functions. 


WITH   THE    REQUISITE   TABLES. 
BY 

HERBERT  L.  RICE,  M.  S., 

Assistant  in  the  Office  of  the  American  Ephemeris, 
AND    Professor  of  astronomy  in  the  Corcoran  Scientific  School,  Washington,  D.C. 


LYNN,     MASS. 

The   Nichols   Press  —  Thos.    P.    Nichols. 

1899. 


'-3  |4- 


Copyright,  1899,  by 

HERBERT    L.     RICE, 

Washington,  D.C. 


PREFACE. 


In  preparing  the  following  treatise  the  author  has  attempted  no  marked 
originality,  either  of  subject  matter  or  method.  Indeed,  sufficient  has  hitherto 
been  written  of  Interpolation,  Quadratures,  etc.,  to  firmly  dissuade  one  from 
such  an  endeavor.  Yet  of  the  numerous  contributions  to  these  allied  subjects, 
there  has  appeared  tlius  far  no  distinct  treatise  covering  the  entire  ground.  As 
a  consequence  the  autlior  has  repeatedly  felt  the  need  of  a  work  which  would 
give — exclusive  of  other  matter  —  a  simple,  practical,  yet  comprehensive  discus- 
sion of  all  that  is  useful  concerning  Differences,  Interpolation,  Tabular  Differ- 
entiation and  Mechanical  Quadrature;  —  a  work,  moreover,  which  would  include 
all  tables  appertaining  to  the  text  which  are  required  by  a  practical  computer. 
To  supply  the  want  thus  conceived,  tlie  autlitu-  offei'S  the  present  volume. 

But  while  viewing  the  matter  in  this  practical  sense,  the  writei'  regards  his 
work  as  no  mere  compilation.  Many  of  the  processes  and  developments  are 
original,  so  far  as  he  is  concerned,  and  possibl}'  altogether  new ;  while  the  same 
remark  applies  to  a  few  of  the  minor  results.  In  fact,  if  adverse  criticism  be 
forthcoming,  it  will  probably  result  largely  from  the  somewhat  unusual  or  indi- 
vidual methods  which  in  many  instances  iiave  been  emploj'ed  in  preference  to 
the  customary  forms  of  analysis.  On '  the  other  hand  the  author  realizes  fully 
the  extent  of  his  indebtedness  to  previous  writers  for  valuable  ideas  and  sug- 
gestions :  and  he  desires  especially  to  mention  the  works  of  Boole,  Chauvenet, 
Encke,  Loomis,  Newcomb,  and  Sawitsch  as  most  valuable  sources  of  informa- 
tion, to  which  frequent  reference  has  been  made. 

Concerning  the  bibliographical  list  at  the  close  of  this  volume  (whicli 
includes  the  foregoing  names),  it  is  but  proper  to  state  that  references  to 
several  of  the  earliest  writers  —  sucli  as  Briggs,  Wallis,  Mouton,  Cotes, 
Stirling,  Mayer,  Walmesley,  Lalande  —  have  purposely  been  omitted  because 
of  the  general  inaccessibility  of  tlieir  works.  As  regards  the  writings  of  the 
present  century,  however,  the  author  believes  that  all  contributions  of  importance 
have  been  included,  and  trusts  that  any  omissions  of  consequence  hereafter 
detected  will  be  regarded  merely  as  oversights. 


IV  PREFACE. 

Special  care  has  been  given  to  the  preparation  and  printing  of  the  tal)les, 
with  the  hope  of  securing  absolute  accuracy.  At  a  considerable  cost  of  lahoi-, 
and  by  wholly  independent  methods,  the  computations  were  all  made  in  dupli- 
cate; and  in  every  case  the  tabular  values  are  true  to  the  nearest  unit  of  the 
last  place.  Though  a  few  of  these  tables  have  appeared  before,  several  are  here 
published  for  the  first  time,  and  it  is  hoped  they  will  prove  useful  to  the 
computer. 

In  conclusion,  the  author  desires  to  express  his  cordial  thanks  and  appre- 
ciation to  Mr.  E.  C.  RuEBSAJt,  of  the  Nautical  Almanac  OfEce,  and  to  Mr.  M.  E. 
PoKTER,  of  the  Naval  Observatory,  for  much  valuable  service  and  many  useful 
suggestions  received  during  the  various  phases  of  preparation  of  this  treatise. 
Feelings  of  gratitude  further  inspire — simple  justice  even  demands  —  a  special 
word  in  commendation  of  the  publishers,  whose  uniform  courtesy,  accuracy  and 
skill  have  done   much  to  enhance  the  general  value  of  the  work. 

H.  L.  R. 

W'a.suixgtox,  D.C,   December,  1899. 


CONTENTS. 


CHAPTEK    I. 


OF      DIFFERENCES. 


Section.  Pajcc. 

1.  General  remarks  concerning  tabular  functions  and    the  construction  of 

mathematical  tables,          .........  1 

2.  Fundamental  definitions  and  notation.      General  schedule  of  functions 

and  differences,          ..........  2 

3.  Method  of  checking  the  numerical  accuracy  of  differences.     Theorem  I,  4 

4.  N  functions  yield  N—n  wth  differences.     Theorem  II,            ...  5 

5.  Effect  of  inverting  a  given  series.     Theorem  III,            ....  5 

6.  Differences  of  two  combined  series.     Theorem  IV,          ....  6 

7.  Irregularities  in  the  differences  of  functions  which  aie  not  mathemati- 

cally exact,        ...........  7 

S.            Detection  of  accidental  errors  by  differences, 9 

9.             Numerical  examples — in  which  only  one  function  requires  correction,  11 

10.  Numerical  examples  —  involving  two  or  three  erroneous  functions,        .  13 

11.  General  properties  of  differences.     Expression  of  z/i"'  in  terms    of   the 

wth  and  higher  derivatives  of  I'^(t); — equation  (4),        .         .         .  ir> 

12.  Determination  of  the  coeiBcients  B,  C,  D,  etc.,  in  equation  (4),     .         .  18 

13.  Remarkable  formal  relation  between  the  expressions  for  //<,"*  and  ^'„,     ■  21 

14.  The  //th  differences  of  any  rational  integral  function  of  the  ii\h  degree 

are  constant.     Theorem  V,         ........  23 

15.  Converse  of  the  foregoing  proposition.     Theorem  VI,     .          .          .          .  24 

16.  17.     Convergency  of  differences.      Ma;/nitude    of  talmlar   Interral    and  rhar- 

acter  of  function,  the  principal  elements  involved.     Numerical  illus- 
trations,    ............  25 

18.  Expression  of  wF'(t),  <o"F"{t),  etc.,  in  terms  of  tabuhar  differences,          .  28 

19.  Change  of  the  argument  interval  from  w  to  mm:  effect  upon  the  mag- 

nitude of  the  successive  differences,         ......  30 

20.  Practical  result  of  the  foregoing  investigation.     Theorem  VII,      .         .  34 

21.  Numerical  example  —  reduction  of  tabular  interval,        ....  34 

22.  Expression  of  any  difference  in  terms  of  tabular  functions,           .          .  35 

23.  Expression  of   any  tabular  function  in  terms  of   F^,  z/^,  J^',  z/^",  etc.,  3(> 
Examples,          ............  38 


VI  CONTEXTS. 


CHAPTEE    II. 


OF      INTERPOLATION. 
Section.  Pai?e. 

24.  Statement  of  the  problem,       .........         40 

25.  Eigorous   jnoof   of   Newton's   Formula,   assuming    that   differences    of 

some  particular  order  are  constant,           ......  41 

26.  Second  demonstration  of  Newton's  Formula,  restricted  as  in  §25,       .  43 

27.  Formula  for  computing  the  interval  n,  .         .         .         .         .         .          .  43 

28.  Example  of  interpolation  by  Nkwton's  Formula,  the  fourth  differences 

being  constant,  ..........  44 

29.  Backward  interpolation  by  Newton's  Formula.     Interpolation  near  the 

end  of  a  series.     Numerical  example,       ......         44 

30.  General  investigation  proving  that  Newton'.s  Formula  is  sensibly  accu- 

rate as  applied  to  series  whose  differences  practically  —  though  not 
absolutely  —  vanish  beyond  the  4th  or  5th  order,  ...         46 

31.  Numerical  example  illustrating  the  foregoing  discussion,       ...         57 

32.  33.     Practical  examples  in  the  use  of  Newton's  Formula,    .         .         .         .         Gl 

34.  Transformations  of  Newton's  Formula.     Modification  of  the  foregoing 

notation    of   differences.     Stirling's  Formula.     Schedule  of  differ- 
ences referring  to  same.     Example,          ......  62 

35.  Backivard  interpolation  by  Stirling's  Formula.     Example,   ...  65 

36.  Further  example  in  the  use  of  Stirling's  Formula,       ....  65 

37.  The  al'jehrak  mean.     Practical  precepts,          ......  66 

38.  Derivation  of  Bessel's  Formula.     Numerical  application,        ...  67 

39.  Second  example  of  interpolation  by  Bessel's  Formula,          ...  68 

40.  Backward  interpolation  by  Bessel's  Formula.     Example,      ...  69 

41.  Property  of  Bessel's  Coefficients,    ........  69 

42.  Comparison    of   the    relative  advantages  of  Newton's,  Stirling's,  and 

Bessel's  Formulae,   ..........         71 

43.  Simple    interpolation.      Magnitude    of    error    arising    from    neglect    of 

second  differences,     ..........         72 

44.  Interpolation  by  means  of  a  corrected  first  difference.     Example,  .         .         73 

45.  Backward     interpolation    by    means     of    a    corrected    first    difference. 

Examples,  ...........         74 

46.  47.     Correction    of    erroneous    tabular    functions    by    direct    interpolation. 

Example,  ............         76 

48.  Systematic    interpolation    of   series.     Reduction  of   a   given  tabular  in- 

terval,       ............         78 

49.  Interpolation  to  halves.     Practical  rule,  ......         SO 

50.  Precepts    for    systematic    interpolation    to    halves.      Schedule    showing 

arrangement  of  quantities.     Numeiical  example,      ....         81 

51.  Derivation  of  general  formulae  for  reducing  the  tabular  interval  from 

o)  to  Htu),  m  being  the  reciprocal  of  a  positive  odd  integer,    .         .         83 


CONTENTS. 


VU 


Section.  Page. 

52.  Systematic  interpolation  to  thirds.     Example, 88 

53.  Systematic  interpolation  to  ffths.     Example,           .....  89 

54.  On   the   best   order   of   performing   successive  interpolations  to  halves, 

thirds,  etc.,         ...........  91 

55.  Interpolation,  with  a  constant  interval  ii,  of  an  entire  series  of  functions. 

Example,  ............  91 

Examples,          ............  94 


CHAPTER   III. 


DEKIVATIVES    OF    TABULAR    FUNCTIONS. 


66.  Concerning   the   close    relation    between  differences  and  differential  co 

efficients,    .         .         .         .         .         .         .         .         . 

57.  Practical  applications  of  formulae  resulting  from  this  relation.     Impor 

tance  of  tabular  derivatives  in  Astronomy,      .... 

58.  Derivation  of  the  required  formulae  in  general  terms,  . 

59.  Formulae  for  computing  derivatives  at  or  near  the  beginning  of  a  series 

Examples,  .......... 

60.  Formulae  apijlicable  at  or  near  the  end  of  a  series.     Examples,   . 

61.  Derivatives    from     Stirling's    Formula.       Rule    for    computing    F'(t) 

Examples,  .......... 

62.  Derivatives  from  Bessel's  Formula.     Simple  expression  for  F'(t-\-^u>) 

Applications  and  examples,       ....... 

63.  Interpolation  by  means  of  tabular  first  derivatives.     Example, 

64.  Application    of   preceding   method  when    second  differences   are  nearly 

constant.     Practical  rule  for  this  case.     Examples, 

65.  Regarding  a  choice  of  formulae  in  any  given  case. 
Examples, 


97 

97 

98 

101 
105 

109 

115 
121 

124 
127 
128 


CHAPTER   IV. 


OF    MECHANICAL    QUADRATURE. 


66.  Statement  of  the  problem.     Important  applications  of  the  method,       .       130 

67.  Derivation  of  formulae  for  single  integration  from  Newton's  Formula. 

The   auxiliary  series    'F.     Schedule   of   functions   and   differences,       131 

68.  Numerical  applications  illustrating  two  of  the  foregoing  formulae,         .       137 

69.  Precepts  for  computing  a  definite   integral  when   either  or  both  limits 

are   other   than   tabular  values  of  the  argument  T.     Necessity  of 
interpolation  in  this  case,         .         ,         .         .         .         .         .         .138 


Vlll  CONTENTS. 

Section.  Page. 

70,  71.  Transformation  and  extension  of  tlie  fundamental  relations  of  §67, 
such  that  integrals  whose  limits  are  non-tahulaf  values  of  T  are 
expressed  directly  in  terms  of  Interjiolated  values  of  '/',  F,  J',  zl",  z/'", 
etc.     Formulae  and  examples,    ........       140 

72.  Formulae  for  single   integration   as  derived  from    Stirlixo's  Formula. 

Schedule  of  functions  and  differences.     Examples,  .         .         .       14G 

73.  Generalization  of  preceding  formulae  to  include  integrals  of  ani/  limits. 

Example, 151 

74.  Formulae  for  single  integration  from   Eessel's  Formula.     Extension  to 

(1711/  limits.     Examples,     .........       153 

75.  Double  integration.     The  conditions  involved,         .....       IGO 

76.  Derivation  of  formulae  for  double  integration  from  Newton's  Formula. 

Introduction   of   the  series  "F.     Schedule  of  functions  and  differ- 
ences.    General  fornndae  and  relations,  .....       IGO 

77.  Value  of   the  frst  integral  at  the  lower  limit.     Introduction  and  defi- 

nition   of   tlie    quantity    //„.       Collection    of    formulae    for    double 
integration  covering  all  possible  cases.     Examples,  .  .  .        IGG 

78.  Derivation    of   formulae    for   double   integration    from    Stirling's    and 

Bessel's    Formulae.      Schedule    referring    to    same.      Precepts    and 
examples,  .         .         .         .         .         .         .         .         .         .         .         .173 

79.  Change  in  value  of  the  double  integral    Y,  due  to  an  arbitrary  change 

in  the  constant  If,    .........         .       188 

Examples, 189 


CHAFTER    V. 
miscellaneous  problems  and  applications. 

80.  Introductory  statement,    ..........        191 

81.  Problem  I.  —  To  find  the    sum    of   the  Ath  powers   of   the  first  ;•  inte- 

gers.    Application  to  >S'=  1^-1-2^  +  3^  + ^-r^       ....       191 

82.  83.     Problem  II.  —  Given  the  series    F_„,  F_^,  F„,  F^,  F„,    etc.,    and  an  as- 

signed value  of  i''„;    to  find    the    corresponding   interval    n.      Two 
solutions.     Examples,         .........       192 

84.  Problem  III.  —  To    solve    any   numerical   ccpiation    containing    but   one 

unknown  quantity.     Example,  .......       195 

85.  Problem  IV. — To  find  the  value   of   the   argument   corresponding  to  a 

maximum  or  minimum  function.     Example,     .         .         .         .         .196 

86.  Problem  V. — Given  a  series    of   values,    F_^,  F_^,  F„,  F^,  F^,    etc.,  of 

some   function    F {T)    analytically  unknown;    to   find    an    approxi- 
mate algebraic  exjiression  for  /'(?').     Examples,     ....  198 

87.  Geometrical  problem,        ..........  200 

88.  Concluding  remarks,         .     ^ 202 

Examples, 203 


CONTENTS. 


IX 


APPENDIX. 


ON    THE    SYMBOLIC    METHOD    OK    DEVKLOPMENT. 


Section. 

89.  Introductory  remarks,       ......... 

90.  Definition  and  operation  of  the  symbols  A,  A'-,  A",  etc., 

91.  Definition  and  operation  of  D,  D^,  D',  etc.,      ..... 

92.  93.  Proof   that   the  foregoing    symbols    of   operation   obey,  in    general,  the 

fundamental  laws  of  algebraic  combination,     .... 

94.  Consideration  of  negative  powers  of  A  and   D,      . 

95.  Remark  concerning  results  established  in  the  preceding  sections, 

96.  Demonstration  of  Theorem  III,        ....... 

97.  Fundamental  relation  between  A  and   D,        .         .         .         .         . 

98.  Expression  of    A,  A^  A',    etc.,   in    terms    of   ascending    powers   of    D, 

Demonstration  of  Theorem  V,  . 

99.  Expression  of  D,  D'-',  D',  etc.,  in  terms  of  ascending  powers  of  A, 

100.  Reduction  of  the  tabular  interval  o).     Expression  of   <?,  a'-',  d',   etc.,  in 

terms  of  ascending  powers  of  A,    . 

101.  Effect  of  the  operator    1  +  A.     Newton's  Formula  of  interpolation, 

102.  Definition  of  the  symbol   of   operation   V.     Its  relation  to  A   and   D, 

103.  Derivation  of  Newton's  Formula  for  backward  interpolation, 

104.  Expression  of   any  difference  in  terms  of   the  given  tabular  functions, 

105.  Derivation    of    the    fundamental    relations    of    mechanical    quadrature 

Single  integration,     ......... 

106.  The  fundamental  formulae  of  double  integration,  .... 


Page. 
205 

205 

206 

206 
208 
209 
209 

210 

210 

2n 

211 

211 
212 
213 
213 

214 
214 


TABLES. 


Table  I.  Newton's  coefficients  of  interpolation, 

Table  II.  Stirling's  coefficients  of  interpolation. 

Table  III.  Bessel's  coefficients  of  interpolation,     . 

Table  IV.  Newton's  coefficients  for  computing  F'(T), 

Table  V.  Stirling's  coefficients  for  computing  F'(T), 

Table  VI.  Bessel's  coefficients  for  computing  F'{T),    . 

Table  VII.  Giving  //:  For  finding  ?i  when  F^  is  given,  . 

Table  VIII.  Coefficients  for  interpolating  by  means  of  tabular  first  derivatives, 


218 
220 
222 
224 
226 
228 
230 
232 


Bibliography, 


233 


Do    ^^ 


CIIAITEII  I, 


OF    DIFFERENCES. 


1.  In  many  applications  of  tlie  exact  sciences,  and  of  Astrononi}' 
in  particular,  it  is  often  necessaiy  to  tabulate  a  series  of  numerical 
values  of  some  quantity  or  function,  corresponding  to  certain  assumed 
values  of  the  element  or  argument  upon  which  the  fiuictional  values 
depend. 

In  the  more  purely  mathematical  tables,  the  function  is  analyti- 
cally known  ;  the  argument  is  then  the  independent  variable  of  the 
given  expression.  The  common  tables  of  logarithms,  trigonometrical 
functions,  squares,  cubes,  and  reciprocals,  are  examples  of  tabular 
functions  of  this  class. 

A  second  and  larger  class  includes  those  functions  which  are  not 
related  analytically  to  the  argument,  but  which  are  either  determined 
directly  by  experiment,  or  based  wholly  oi'  partly  upon  observation. 
The  final  results  are  usually  obtained  from  the  fundamental  obser- 
vations by  suitable  mathematical  transformations  or  reductions,  which 
frequently  include  the  process  of  adjustment  known  as  the  method  of 
least-squares.  Empirical  values  are  also  occasionally  introduced  in  the 
development  of  functions  of  this  class,  to  supply  some  theoretical 
deficiency. 

In  the  great  majority  of  such  cases,  the  finie  is  the  argument  of  the 
tabi;lated  function.  This  is  particularly  the  case  in  astronomical  tables. 
Thiis  the  Nautical  Almanac  gives  the  right-ascensions  and  declinations 
of  the  sun  and  the  planets  for  every  Greenwich  mean  noon  ;  in  the 
case  of  the  moon,  these  coordinates  are  given  for  every  hour,  because 
of  the  I'apid  motion  of  our  satellite.  The  moon's  horizontal  parallax 
is  tabulated  for  every   twelve  hours  ;    the  sun's  for  every  ten  days. 

In  like  manner,  the    readings    of  the    barometer   and    thermometer 


2i  THE    TIIEORV    AM)    I'llACTICE    OF    INTERPOLATION. 

are  recorded  for  certain  hours  of  tlie  day,  and  lln'reCore  may  be  i-egarded 
as  functions  of  tlie  time.  The  velocity  of  tlic  wind,  tlu'  heit;lit  of  tide- 
water, the  correction  and  rate  of  a  clock,  are  fui'tiier  instances  of  a 
large  number  of  jihysical  quantities  whicli  ai-e  tabulated  as  functions 
of  the  time. 

As  examples  of  tabular  functions  of  the  physical  or  observational 
kind,  whose  arguments  are  elements  other  than  tlie  time,  we  may 
mention  : 

(«)  The  force  of  gravity  (determined  by  pendulum  experiments), 
as  a  function  of  the  latitude  ; 

(&)  The  atmospheric  pressure  (determined  by  the  barometer),  as 
a  function  of  the  altitude  ; 

((■)  The  angle  of  refraction  in  a  particular  substance,  as  a  function 
of  the  angle  of  incidence. 

Although  differing  thus  fundamentally  in  the  character  of  their 
respective  functions,  all  mathematical  tables  are  alike  in  giving  the 
numerical  values  of  the  functions  for  certain  assumed  values  of  the 
argument,  so  chosen  that  intermediate  values  of  the  function  may 
readily  be  derived  by  the  process  of  interpolation.  For  this  purpose 
it  is  convenient,  though  not  essential,  to  have  the  assumed  argument 
values  proceed  according  to  some  law  ;  and  since  as  a  rule  the  greatest 
simplicity  is  attained  whei-e  the  argument  varies  uniformly,  it  is  nearly 
always  so  taken.  The  interval  of  the  argument  is  decided  in  general 
by  the  rapidity  with  which  the  given  function  varies. 

We  shall  assume  throughout  these  pages  that  the  given  values  of 
the  argument  are  equidistant. 

The  present  chapter  will  be  devoted  to  the  subject  of  diferences, 
as  defined  below.  The  student  should  become  thoroughly  and  practi- 
cally familiar  with  this  fundamental  jjortion  of  the  work  before  entering 
upon  the  chapters  that  follow. 

2.  Definitions  and  Notation. —  If  wc  have  given  a  series  of 
quantities  proceeding  according  to  any  law,  and  take  the  difference 
of  every  two  consecutive  terms,  we  obtain  a  series  of  values  called 
the  first  order  of  differences,  or  briefly,  first  differences. 


THK  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


3 


If  we  difference  the  first  differences  in  tlie  same  Dianner,  we 
form  a  new  series  called  second  dlfferenees.  The  process  may  be  con- 
tinued, if  necessary,  so  long  as  any  differences  remain. 

We  shall  apply  this  process  of  differencing  to  the  tabular  values 
of  functions  given  for  equidistant  values  of  the  argument. 

Let   T  designate  the  argvunent;    w,  its  interval;    F{T),  ov  simply 

F,  the  function  ;     t,t-\-w,t-{-  2a),  t  -\-  Sw, ,     the  given  values 

of  T;     Fa,  F,,  F.2,  F^, ,     the  corresponding  values  of  F(T); 

J',  A",  J'",  J",  .  .  .  .  ,  the  successive  orders  of  differences.  The  arrange- 
ment is  then  shown  in  the  following  schedule  : 


Argument 

Function 

lstDiff.!2dDiff. 

3d  Diff. 

J"' 

4th  Diff. 

.5th  Diff.  :6th  Diff. 

T 

F(T) 

J' 

J" 

Jiv 

Jv 

jvl 

f 

F. 

t  +  «, 

F. 

"o 

*o 

;;  +  2<o 

F„ 

«i 

^ 

«o 

cZ„ 

f  +3w 

F, 

a„ 

h 

"i 

d. 

«o 

./■o 

^  +  4«, 

F, 

«3 

K 

Cl 

d. 

ei 

t  +  5<i> 

F, 

at 

K 

"3 

t  +  6a) 

F, 

«5 

where  a^^F,—F„,  n^^=^  F^ — i^i,  .  .  .  ;  &o  =  «i — «o,  ^>i  =  «.,— «i,  •  •  •; 
Cf,^bi  —  b„,  Ci^Ih — />, ,  .  .  .  ;     and  so  on. 

We  shall  also  find  it  convenient  to  represent     a^,  a^,  a.,,  ....     by 

J/,  4',  J,',  ....  ,    respectively;    \,  &i,  Z>.,,   ....    by  4,",  4",  4"> >  etc., 

Thus,  generally,  ./;"'  denotes  the  (.■^-j-l)"'  value  in  the  column  of  «"' 
differences. 

As  an  example,  we  tabulate  and  difference  several  successive 
values  of    F i^T)  =  T'— 10 T-—  20,     thus  : 


THE  THEORY  AND  PRACTICE  OF  IKTERPOLATION. 


T 

F(T) 

J' 

J" 

J'" 

Jiv 

Jv 

0 

-   20 

—  9 

1 

-  29 

-  15 

-  6 

+  36 

2 

-  44 

+  15 

+  30 

+  60 

+  24 

0 

3 

-  29 

+  105 

+  90 

+  84 

+  24 

0 

4 

+  7f) 

+  279 

+  174 

+  108 

+  24 

5 
(i 

+  355 

+  91G 

+  561 

+  282 

The  difterences  are  in  all  cases  formed  by  subtracting  (algebrai- 
cally) downwards,  as  in  the  above  examijles.  It  will  be  noted  that 
the  even  differences  (j",  J",  .  .  .  .  )  always  fall  on  the  same  lines 
with  the  argument  and  function,  Avhile  the  odd  differences  ( J',  //'",  /P,  .  .  .) 
lie  between  the  lines. 

3.  Method  of  Checking  the  Numerical  Accuracy  of  the  Differ- 
ences. —  If,  in  the  numerical  example  of  the  last  section,  we  take  the 
algebraic  sum  of  the  six  given  values  of  J',   we  find 

_9  _  15  4-  15  +  105  +  279  +  561    =    +936 

Subtracting  the  first  value  of  F{T)  from  the  last,  we  have 

+  916  _  (-20)   =   +936 

which  agrees  with  the  first  result. 
Again,  in  like  manner,  we  find 

J,"'+Ji"'+J2"'  =    +36  +  60  +  84  =    +180   =    +174 -(-6)   =  4"--^o" 

These  relations  may  be  expressed  generally  as  follows  : 
Theorem  I. —  The  algehraic  sum  of  any     s     consecutive  values  of 

/]("\     is   equal   to   the   last,   minus   the  first,   of  the     s-\-l     consecutive 

j(„-i)     terms  used  in  forming  the     s     values  of  /P"K 

To  prove  this  proposition,  let  the  differences  be  as  below  : 

//'»-"  :    ;,,    h,   h, //._,    r,,    h,^, 

^'"'        :  /'^      h     '^, ^.-1   f^. 


THE    TIIEOKY    AND    PRACTICE   OF    IKTERPOLATION.  5 

Then,  from  the  (h'finition  ol"  differences,  we  have 

A-,  =  A,-A,,         k,  =  l,,-h,,         ,         k,_,  =  h,-h,_,,         k,  =  h,^^-h. 

Hence,  by  additiuii,  we  find 

A-,  +  k,  +  k,  + +  k,_,  +  /■.  =  h,^,  -  A, 

which  is  the  algebraic  statement  of  Theoi-em  I.  This  theorem  may 
obvionsly  be  applied  as  an  independent  check  upon  the  numerical 
accuracy  of  the  differencing. 

4.  Theorem  II.  —  //"  the  differences  of  N  values  of  F{T) 
are  taken,  N — n  values  of  z/<"*  are  derived ;  it  being  assumed 
that     jV>n. 

For,  iV  functions  evidently  yield  JV — 1  vahies  of  y,  JST- — 2  values 
of  J",  iV— 3  values  of  j"',  etc.  ;  hence  JV  values  of  F{T)  yield  iV— ?i 
values  of  j<'". 

5.  Inversion  of  a  Series  of  Functions.  —  It  is  sometimes  necessary 
or  convenient  to  invert  a  given  column  of  functions,  thus  bringing  the 
last  value  into  the  position  of  the  first,  the  next  to  the  last  into  the 
position  of  the  second,  etc.  For  example,  let  us  invert  the  series 
given  in  §2,  and  observe  the  effect  of  this  inversion  upon  the  differ- 
ences.    Thus  we  find  : 


T 

F(T)           J' 

J" 

J'"      Jiv     Jv 

6 
5 
4 
3 
2 
1 
0 

+  916 
+365 
+  76 

-  29 

-  44 

-  29 

-  20 

-661 
-279 
-105 
-  15 
+  15 
+  9 

+  282 
+  174 
+  90 
+  30 
-  6 

-108 

-  84 

-  60 

-  36 

+  24 
+  24 
+  24 

0 
0 

Comparing  this  table  with  the  original,  we  first  observe  that  each 
column  of  differences  is  inverted,  like  the  column  of  functions  itself. 
Further,  having  regard  to  signs,  we  see  that  the  first  and  third  differ- 
ences (the  odd  orders)  have  changed  signs  throughout  ;  while  J"  and 
//'^  (the  even  orders)   remain  unaltered  in  sign. 


6 


TlIK    TUEOKY    AM)    PRACTICE    OF    IXTKKPOLATION. 


To    prove    that    such    an    eifect    is  true  generally,  we  consider  the 
two  series  below,  the  second  series  being  an  invi'ision  of  tlic   lirst  : 


F{T) 

J' 

J" 

J'" 

Jiv 

F(T) 

J' 

J" 

J'" 

Jlv 

«o 

*o 

F. 
F, 

.  «o 

/80 

F, 

a. 

^ 

'•0 

d„ 

F, 

«1 

/8, 

Vo 

K 

Fs 

ffj 

K 

''i 

d. 

F„ 

P'l 

Vi 

s, 

F. 

F:, 

"3 

"4 

K 

Cs 

^0 

«3 
«4 

A 

y-.- 

Comparing-  the  first  differences,  we  find 


«i   =   F,  -  F,  =    _(//_/.;)   = 


—  a.. 


Hence,  for  the  second  differences,  we  obtain 

Po    =    "1  —  «o    =     —0;!  -  (-«4)     =    "4  -  «:! 
/3|    =    «2  —  «i    =    — ".J  —  ( —  a.,)    =    a.^  —  a„ 


=    /, 


Thus,  the  inversion  of  the  functions  inverts  J',  and  changes  its  signs 
throughout  ;  whereas  j"  is  inverted,  but  does  not  change  in  sign, 
Further,  since  j"'  and  .7'^  have  the  same  relation  to  J",  that  j'  and  J" 
have  to  F{T),  it  is  manifest  that  J>"  inverts  and  changes  signs, 
Avhile  J'^  inverts  with  signs  unaltered.  Extending  this  i-easoning,  we 
have  the  following  proposition  : 

Theokem  III.  —  Inverting  a  series  of  functiotis  inverts  each  column 
of  differences  and  cha7iges  the  signs  of  the  odd  orders  (//',  J'",  z/^, .  .  .  .), 
uihile  the  signs  of  the  even  orders     (J",  zl",  .  .  .  .)   remain  unchanged. 

In  pi-actice  it  is  seldom  necessary  to  re-tabulate  tlic  function  in 
the  inverted  order,  since  we  may  I'cadily  conceive  the  inversion  to  be 
made,  merely  allowing  for  the  changes  of  sign  in     J',  J'",  J",  etc. 

G.  Theorem  IV.  —  The  v"'  differences  of  the  sitnis  of  two  series 
of  functions  are  equal  to  the  sums  of  the  corresiwnding  n"'  differences 
of  the  two  component  series. 


THE    THEORY   AND    PRACTICE    OF    INTERPOLATION'.  7 

To  prove  genenilly,  Ic-f      I'\.  F^,J'\, ,     and    f„,f^,f,,  .  .  .  .  , 

denote  tlie  two  series  of  i'uiictioiis  ;  then  the  sums  of  the  two  series 
will  be  K-\-fu,  i^i-f:/i,  P'-i-i-f,-  •  ■  •  •  Al.so,  let  us  desig-uate  the  first 
differences  of  these  three  series  by  ./',  8',  and  D',  respectively  ;  their 
values  are  hence  as  follows  : 


J' 


F,-F, 


8' 

A-L 
.t\~-A 


D' 


w +/,)-(/•; +./;,) 
(/';+/=)-(/';+/,) 


We  therefoi'e  have 


A'  =  (^\+A)  -  (^0+./;) 


i'\+A  -  K-fo  =  it'\-K)  +  (f-fo) 

^\+f,  -  ^;-/;  =  {F,-F,)  +  (/,-r\) 


J/ +  8/ 


These  relations  prove  the  theorem  directly  for  /i  :=  1  ;  jjut  since 
the  second  differences  are  formed  fi-om  the  first  difierences  in  the  same 
manner  that  the  latter  are  derived  from  the  given  functions,  the  theorem 
is  also  true  for  n  =  2.  Similarly  with  the  following  differences,  each 
order  being  the  first  difference  of  the  order  just  jireceding.  Hence 
the  theorem  is  true  generally. 

As  an  example  we  write  : 


F 

J' 

J" 

J'" 

-  5 

-  4 
+  9 
+  40 
+  95 

+  1 
+  13 
+  31 

+  55 

+  12 
+  18 
+  24 

+  6 
+  6 

/ 

(5' 

(5" 

,i"' 

+  14 
+  1G 
+  19 
+  19 
+  13 

+  2 

+  3 

0 

-6 

+  1 

-3 
-G 

-4 
-3 

F+f 

D' 

D" 

D'" 

+  9 
+  12 
+  28 
+  59 
+  108 

+  3 
+  16 
+  31 

+  49 

+  13 

+  15 
+  18 

+  2 
+  3 

It  will  be  observed  that  the  values  of  D',  D"  and  D"  are  in 
accord  with  the  theorem. 

7.  Irregularities  in  thr  Differences.  —  In  the  numerical  example 
of  §2,  the  values  of  J"  are  all  zero.  In  such  a  case,  we  say  that  the 
differences  are  perfectly  smooth  or  regular.     In  practice,  however,  the 


8 


THE    THEORY    AND    PKACTICE    OF    IKTEKl'OLATION. 


differences  frequently  exhibit  a  .small  degree  of  irregularity,  owing  to 
the  omission  of  decimals  in  the  approximate  values  of  the  functions 
employed.  As  an  example,  we  take  the  following  \alues  of  T*,  true  to 
the  neai-est  unit  of  the  second  decimal  : 


r 

F{T)^T* 

J' 

J" 

J'" 

Jlv 

2.0 
2.1 
2.2 
2.3 
2.4 
2.5 
2.6 
2.7 
2.8 

16.00 
19.45 
23.43 
27.98 
33.18 
39.06 
45.70 
53.14 
61.47 

+  3.45 
3.98 
4.55 
5.20 
5.88 
6.64 
7.44 

+  8.33 

+  0.53 
.57 
.65 
.68 
.76 
.80 

+  0.89 

+  0.04 
.08 
.03 
.08 
.04 

+  0.09 

+  0.04 

-  .05 
+    .05 

-  .04 

+  0.05 

That  the  irregularity  here  manifest  in  the  outer  differences  is  due 
to  the  fact  that  the  tabular  values  are  only  approximate  (not  the 
true  mathematical  values  of  the  function),  may  easily  be  shown  l)y 
Theorem  IV,  thus  :  let 

F  denote  the  true  value  of  the  function  ; 

F,  its  approximate  value  as  above  ; 

/  =  F—  F,  the  difference  of  these  values. 


Then,  since  F  is  given  to  the  nearest  nnit  of  the  second  place, 
/  may  have  any  value  from  — 0.5  to  -j-O.o,  in  terms  of  the  same  unit. 
Moreovei',  the  values  of  /  do  not  follow  any  law  of  progression,  but 
proceed  at  random,  with  arbitrary  changes  of  sign.  Hence,  the  differ- 
ences of  /  will  be  irregular.  The  differences  of  F  must  proceed  regu- 
larly, however,  since  F  is  the  true  mathematical  value  of  a  continuous 
function.  Now,  since  _F=  F-\-f\  it  follows  from  Theorem  IV  that 
the  differences  of  F  must  equal  tlie  sums  of  the  corresponding  dif- 
erences  of  F  and  f ;  therefore,  the  differences  of  F  must  contain  just 
such  irregularities  as  are  inevitable  in  the  differences  of  f. 

To  illustrate  this  principle,  we  tabulate  below  the  values  of  F, 
along  with  the  given  series,  F ;  Avhcnce  _/'  follows,  in  units  of  the 
second  decimal,  and  also  the  differences  of/  to  the  fourth  order  : 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION". 


r 

F{T) 

F{T) 

f=F-F 

J' 

J" 

J'" 

Jlv 

2.0 
2.1 
2.2 
2.3 
2.4 
2.5 
2.6 
2.7 
2.8 

16.00,00 
19.44,81 
23.42,56 
27.98,41 
33.17,76 
39.06,25 
45.69,76 
53.14,41 
61.46,56 

16.00 
19.45 
23.43 
27.98 
33.18 
39.0(; 
45.70 
53.14 
61.47 

0.00 
+  0.19 
+  0.44 
-0.41 
+  0.24 
-0.25 
+  0.24 
-0.41 
+  0.44 

+  0.19 

+  0.25 
-0.85 
+  0.65 
-0.49 
+  0.49 
-0.65 
+  0.85 

+  0.06 
-1.10 
+  1..-.0 
-1.14 
+  0.98 
-1.14 
+  1..-.0 

-1.10 
+  2.60 
-2.64 
+  2.12 
-2.12 

+  2.(;4 

■ 

+  3.76 
-5.24 
+  4.76 
-4.24 

+  4.76 

We  now  bring  together,  iroxn  the  above  tables,  the  fourth  differ- 
ences of  i'''  and /'  denoting  these  quantities  by  (.7'^)/''  and  {J")f,  resj^ec- 
tively.  The  fourth  differences  of  F  then  follow,  since  we  have  shown 
that     {J")F  =  (J''')F+  (J'")/;     thus  we  form  the  table  below  : 


(J'v)F 

(J"')/ 

(Jiv)^ 

+  0.04 
-0.05 
+  0.05 
-0.04 

+  0.05 

4  0.03,76 
-0.0.5,24 
+  0,04,76 
-0.04,24 
+  0.04,76 

+  0.0024 
+  0.0024 
+  0.0024 
+  0.0024 
+  0.0024 

It  will  be  observed  that  the  fourth  differences  of  F'(T)  ai-e 
absolutely  uniform,  —  that  is,  the  ii'regularities  in  (j'^)/'  and  (./'^')/ ex- 
actly correspond,  or  Ijalance.  The  slight  irregularity  in  the  outer 
differences  of  the  series  F{T)  is  therefore  due  entirely  to  the  omis- 
sion of  decimals,  since  it  wholly  disappears  when  we  employ  the  true 
mathematical  values,  F(T). 

As  a  valual)le  exercise,  the  student  should  now  difference  the 
function  F  directly,  and  Hud  the  foui'th  differences  exactly  as  above 
deduced. 

8.  Detection  of  Accidental  Errors. — We  have  just  seen  hoAv  a 
slight  deviation  from  the  true  value  of  a  tabular  function  will  mani- 
fest itself  by  means  of  irregularities  in  the  differences.  If,  then,  some 
one  value  of  a  series  is  in  error  by  an  ajjpreciable  quantity,  an  in- 
spection of  the  differences  will  indicate  definitely  the  location  and 
magnitude  of  the  error  sought. 


10 


THE    THEOKY    AND   PRACTICE    OF    INTERPOLATION. 


To  investigate  tlie  principle  that  underlies  the  method,  let 

F     F      F      F      F      F 

denote  the  correct  values  of  any  function  F' {T)  (tabulated  for  equi- 
distant vahies  of  T),  and  let  the  diftei'ences  be  as  shown  in  the 
schedule  below  : 


F{T) 

J' 

J" 

J'" 

Jlv 

Jv 

F, 
F, 

a„ 

/> 

F., 

«i 

''. 

C,l 

'4 

F, 
F. 
F. 

"2 

k. 

C] 

(/, 

<'o 

a.. 

h., 

f„ 

(i 

e. 

"4 

1^4 

'3 

(/, 

«-2 

F, 
F. 

h, 

''4 

''4 

'^4 

F, 
F, 
F^o 
Fu 

a- 
(ho 

h. 

'■(1 

'4 

(/- 

F,, 

"u 

Let  us  now  assume  that  some  one  function,  say  F^,  is  in  eri'or 
by  the  quantity  e,  so  that  i^g-j-e  is  tabiilated  in  place  of  the  true 
value  F^ ;  the  differences  of  the  incorrect  sci'ies  will  therefore  be  found 
as  follows  : 


F(T)+t 

J' 

J" 

J"' 

Jiv 

Jv 

Fo 

F, 

«„ 

^'0 

F, 

«i 

'\ 

c'„ 

F. 
F, 

F, 

F,  +  ^ 

F, 

Fs 

F, 

F:„ 

Fu 

"2 

«3 

«,  +  £ 

"7 

«8 
O9 

h.. 
64+   € 

^1(1 

'■1 

Cj-3€ 
6-5  + 3e 

C9 

'^1 

ff2+« 

(/,-4£ 

d,  +  6£ 

d.-4£ 

Fn 

"u 

Now,  because  the  differences  of  the  correct  table  contain  no 
irregularities,  we  see  that  the  differences  of  the  incorrect  table  consist 
of  series  of  regular  values,  to  which  are  alternately  added  and  sub- 
tracted the  terms  in  e,  shown  in  the  above  schedule.  The  law  oi' 
progression  and  increase  in  the  coetiicients  of  e,  along    the   successive 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


11 


orders  of  differences,  is  easily  seen  to  he  that  of  the  hinomial  coef- 
ficients, with  alternate  signs.  Hence,  in  practice,  we  have  only  to 
carry  the  difterencing'  to  that  order  at  which  the  differences  of  the 
correct  functions  would  vanish,  or  scnsihly  so  ;  the  location  and  mag- 
nitude of  the  error  will  then  be  clearly  shown  by  a  succession  of  -|- 
and  —  terms,  following  the  binomial  law. 

Thus,  if  the  values  of  J'  vanish  in  the  correct  table  above,  the 
fifth  differences  of  the  ineori'ect  sei'ies  will  be  0,  -{-e,  — 5e,  -(-lOe, 
—  lOe,  -|-5e,  — e,  0;  the  initial  value,  -|-e,  is  therefore  the  error  sought, 
both  as  to  magnitude  and  sign.  The  required  function  is  found  by 
tracing  backwards  and  downwai'ds  along  the  line  of  heavy  type  from 
gj-j-e  to  J^\-\-e,  which  is  the  incorrect  function  ;  and  since  the  cor- 
rection is  the  negative  of  the  error,  we  have  {I^\-\-e) — e,  or  i^;,  for 
the  true  value  of  .the  function  in  question. 

9.  We  shall  now  consider  several  examples,  in  order  that  the 
process  may  be  fully  understood. 

Example  I.  —  Find  the  error  in  the  following  table  of  F{T)  =  T^: 


T 

F(T)  =  T^ 

J' 

J" 

Jill 

Jiv 

c 

Jiv+C 

1 

2 

3 

1 

8 
27 

+     7 
19 

'i'7 

+  12 
18 

+   6 
+   6 

-  4 

+  36 
-24 
+  16 

+   6 

0 

0 

4 

64 

61 

81 

137 

169 
217 

+  271 

24 

-10 

+  10 

0 

5 

125 

20 

+  40 

-40 

0 

6 

206 

56 

-60 

+  «;o 

0 

7 
8 
9 

343 
512 

729 

32 

48 

+  54 

+  40 
-10 

-40 
+  10 

0 
0 

10 

1000 

The  differencing  is  continued  until  we  find  a  complete  alternation 
of  signs,  as  in  J'^'.  Now  the  binomial  coefficients  of  the  fourth  order 
are  1,  4,  6,  4,  1 ;  it  is  also  seen  that  the  values  of  J'^  are  just  these 
numbers  multiplied  by  10.  Hence,  an  error  of  10  units  exists  some- 
where in  the  function  F ;  its  location  is  easily  determined  by  ti-aeing 
backwards  and  downwards  along  the  line  of  — 10,  — 4,  +20,  4-81,  to 
the  number  206,  which  is  the  quantity  sought.  The  required  function 
is  also  found  by  tracing  backwards  and  upwards  along  the  line  of 
— 10,  -j-lG,  +32,  -f-loT,  to  206  ;  in  practice,  both  lines  should  be 
followed,  to  guard  against  mistake. 


12 


THE    THEORY    AND    PRACTICE    OF    INTEHl'OI.ATIOX. 


Finally,  the  number  l!()()  is  too  sDiall  by  10  unit.s,  since  the  sign 
of  the  ei"i-oi"  is  shown  by  the  leading  or  initial  value  of  the  binomial 
series  in  y,  namely,  — 10.  A  correction  of  -|-10  is  therefore  to  be 
applied    to    the  incoi-rcct  function,  giving  216  foi-  its  true  valne. 

In  the  column  c,  following  J'^  in  the  al)ove  table,  are  given 
the  corrections  to  j'^,  dne  to  the  correction  of  -j-lO  to  the  function. 
The  column  J"-\-c  therefore  gives  tiie  1th  ditferences  of  the  true  or 
corrected  sei'ies.  It  is  alwa3^s  well  to  re-ditference  the  sei'ies  aftei-  a 
correction  has  been  applied,  to  check  the  accuracy  of  the  work. 

Example  II.  —  Find  the  error  in  the  following  table  of  logai'ithms: 


T 

logT 

A> 

J" 

J'" 

c 

J"'+c 

45 
50 
65 
60 
65 
70 
75 
80 

1.6532 
1.6990 
1.7404 

1.7787 

1.8129 
1.8451 
1.8751 
1.9031 

+  458 
414 
383 
342 

322 

300 

+  280 

-44 

31 

41 

20 

22 
-20 

+  13 

-10 

+  21 

-  2 

+   2 

—   5 
+  15 
-15 
+   5 

+  8 
5 
6 

+  2 

The  third  differences  are  here  sufficient  to  point  out  the  erroi'  ; 
the  correction  given  under  c  appears  to  improve  J'"  in  the  best  man- 
ner, thus  indicating  that  log  60  should  be  1.7782  instead  of  1.7787. 
It  will  Ije  observed  that  a  correction  oi"  — (5  is  nearly  as  efficient  as 
— .3  in  the  above  case,  and  that  — 5.5  is  better  than  either ;  this  is 
becau.se  the  value  of  log  GO  to  five  places  is  1.77815. 

Example  III.  —  Correct  the  error  in  tiie  following  ephemeris  of 
the  moon's  latitude  : 


Date 
1898 

Moon's  Lat. 
/3 

J' 

J" 

J'" 

Jiv 

Jv 

c 

A^+c 

May  8.5 
9.0 
9.5 
10.0 
10.5 
11.0 
11.5 
12.0 
12.5 

-1  59  54.2 
1  22  44.2 

0  44  27.0 
-0     5  45.3 
+  0  32  39.9 

1  10  23.4 

1  46  12.4 

2  20  14.7 
+  2  51  51.2 

/            II 

+  37  10.0 
38  17  2 
38  41.7 
38  25.2 
37  43.5 
35  49.0 
34     2.3 

+  31  36.5 

+  1     7.2 
+  0  24.5 
-0  16.5 

0  41.7 

1  54.5 

146.7 

-2  25.8 

II 

-42.7 
41.0 

25.2 

-72.8 
+   7.8 
-39.1 

It 

+   1.7 

+  15.8 
-47.6 
+  80.6 
-46.9 

•     •     ■    • 

II 

+  14.1 

-    63.4 

+  128.2 
-127.5 

ir 

-   12.8 
+   64.0 
-128.0 

+  128.0 

+  1.3 
0.6 
0.2 

+  0.5 

.    .    . 

In    this    example    the    error    is    readily  indicated    in    .d",   for    Avhich 
order   the   binomial   coefficients   are    1,  5,  10,  10,  5,  1.     Although   but 


THE    TIIKORY    AND    PRACTICE    OF    INTERPOLATION. 


13 


four  values  of  r  are  available,  these  are  here  sufficient.  A  slig-ht 
inspection  shows  that  a  correction  of  — 13".(),  as  applied  to  the  latitude 
for  May  11.0,  will  very  nearly  serve  the  purpose;  — 13".()  Ijeiug  a  trifle 
too  great  numerically,  we  soon  find  by  trial  that  — 12".8  produces  the 
best  result.  Hence,  the  moon's  latitude  for  May  11.0  should  read, 
+1°  10'  10".6. 

10.  Correction  of  Errors  when  More  than  One  Function  is 
Affected.  —  Thus  far  we  have  considered  examjDles  of  an  error  in  one 
function  only.  When  two  or  more  consecutive  or  neigh])oring-  vahies 
are  in  error,  the  pi'oblem  of  correction  becomes  moi'c  complicated  and 
difl^cult.  It  may  even  become  indeterminate  in  some  cases,  since  only 
accidental  errors  can  be  detected  by  the  differences.  Several  succes- 
sive functions,  and  possibly  all,  may  contain  systematic  errors  which 
do  not  affect  the  regularity  of  the  differences. 

In  general,  the  correction  of  a  group  of  errors  by  diffei-ences  may 
be  considered  practicable  only  when  the  law  of  the  function  is  not 
obscured  or  altered  l)y  the  presence  of  those  eri-ors.  More  definitely, 
the  method  may  be  regarded  as  available  in  the  case  of  two  or  per- 
haps thi-ee  neighboring  functions,  provided  the  errors  be  accidental  in 
character,  and  of  suiticient  magnitude  to  produce  a  distinct  and  defini- 
tive irregularity  in  the  differences. 

Example  I.  —  Correct  the  errors  in  the  foUoAving  tabulation  of 
F{T)  =  2T'— 25T— 40  : 


T 

F{T) 

J' 

J" 

J'" 

Cl 

J"'+e, 

c. 

J"'+Ci+C-2 

-4 
3 
2 

-J 

+  1 
2 

3 
4 
5 
6 

7 
+  8 

-  68 
^  19 

-  6 

-  17 
40 
63 
79 
61 

-  4 
+  85 

242 

471 

+  784 

+  49 
+  13 

-  11 
23 
23 

-  16 
+  IS 

57 

89 

157 

229 

+  313 

-36 
-  24 
-12 

0 
+  7 
34 
39 
32 
68 
72 
+  84 

+  12 

12 

12 

7 

27 

+  5 

( 

+  36 

4 

+  12 

+  5 
-15 
+  15 

—  5 

+  12 

12 

12 

12 

12 

+  20 

-12 

+  36 

4 

+  12 

-   8 
+  24 
-24 

+  S 

+  12 
12 
12 
12 
12 
12 
12 
12 
12 

+  12 

We    carry  the  difterences    to    the    third    order,    and    note   that    the 
first   three    values    of    /'"  are    constant,    and  equal   to  -|-12  ;    hence,   in 


u 


THE    TIlKOliV    AXr»    PKACTIOE    OF    INTERPOLATION. 


column  c, ,  we  place  tlu'  convction  of  -j-").  Thi.'<  gives  a  corrected 
series  for  J'",  shown  under  ./'"  +  (•,.  The  latter  column  clearly  indicates 
a  correction  of  — 8,  as  applied  in  Cj ;  this  gives  a  final  corrected  column 
of  third  differences,  witli  llic  constant  value  of  -\-V2.  Hence,  ihe 
value  F(7')  for  7'=+L\  should  read  —74  in.stead  of  —7!);  for 
T  =  -j-^'    ^^'^  should  have  — 12  instead  of  — 4. 

Example  II.  —  Correct    the    errors    which    occur    in   the  following 
ephemeris  of  the  sun's  declination  : 


Date 

Sun's  Dec!. 

1S98 

8 

Jan.  28 

-18    6  34.7 

30 

17  34    4.0 

Feb.    1 

17    0  19.0 

3 

16  25  22.9 

5 

15  49  18.8 

7 

15  12    6.6 

9 

14  33  54,0 

11 

13  54  52.8 

13 

13  14  45.0 

15 

12  33  48.1 

17 

11  52    2.4 

19 

-11    9  31.4 

+  32 

30.7 

33 

45.0 

34 

56.1 

36 

4.1 

37 

12  2 

38 

12.6 

39 

1.2 

40 

7.8 

40 

56.9 

41 

45.7 

+  42 

31.0 

J" 

J'" 

fl  *    C-2 

J"'+c,+c„ 

fa 

J"'+Ci 

It 

+  74.3 
71.1 
68.0 
68.1 
60.4 
48.6 
66.6 
49.1 
48.8 

+  45.3 

-   3.2 

II 

-  3.2 

II 

-3.2 

-  3.1 

-    3.1 

3.1 

+   0.1 

-3.2 

-   3.1 

3.1 

1.1 

+  9.6 

+   1.9 

-  5.1 

3.2 

-11.8 

-9.6  +  3.0 

-18.4 

+  15.3 

3.1 

+  1S.0 

+  3.2-9.0 

+  12.2 

-15.3 

3.1 

-17.5 

+  9.0 

-  8.5 

+   5.1 

3.4 

-  0.3 

-  3.5 

-3.0 

-  3.3 

-  3.5 

3.3 
-3.5 

In  this  case,  the  first,  second,  and  last  values  of  A"'  are  — 3.2, 
— 3.1  and  — 3.."),  respectively,  thus  indicating  a  decided  uniform  tend- 
ency in  the  third  differences.  The  first  function  in  ei-ror  is  clearly 
the  value  for  Feb.  7,  and  the  last,  that  for  Feb.  11.  There  may  be  an 
uncertainty  of  a  unit  or  two  in  the  values  of  their  corrections  at  the 
outset;  a  few  trials,  however,  will  indicate  that  — 3.2  is  the  best  value 
to  apply  to  -|-0.1  in  j'",  and  -|-3.0  to  the  term  — 11.8.  By  means  of 
these  corrections,  the  first  three  and  the  last  two  values  of  .;'"  are 
brought  into  pi-actical  uniformity.  In  the  column  c,  &  Cg  are  given  the 
corrections  to  J'",  according  to  the  binomial  numbers,  1,  3,  3,  1.  In 
the  next  column,  the  sum  J"'  +  i\  +  r.^  i.s  written,  which  evidently  requires 
a  third  correction,  tabulated  under  Cg . 

The  differences  are  now  sufficiently  smooth.  Since  c^  coi-responds 
to  a  correction  of  — 5".l  to  8  for  Feb.  9,  we  conclude  that  the  correct 
values  of  8  for  Feb.  7,  9,  and  11,  should  read,  —15°  12'  9".8, 
—14°  33'  59".l,  and  —13°  .'34'  49".8,  respectively. 

It    occasionally    happens    that    some    ordei-    of    difference    clearly 


THE    TIIEOIJV    AXn    PUACTK'F,    OF    INTERPOLATION. 


15 


indicates  a    correction   coiTesponding    to    tlic    hiiioinial    cocfficifnts  of  a 


lower  order  tlian   tliat    of  tlic  differenrc 


question.     '^IMiis  means  tlie 


existence  of  an  eiror  in  some  earlier  oi'der  of  dljf'crrvce,  rather  than 
an  error  in  the  cohnnn  of  functions.  For  example,  if  p  i-equires  a 
correction  of  the  order  1,  3,  .'{,  1,  it  follows  that  an  error  exists  in 
/I",  since  ,P  is  the  third  difl'erence  of  /I".  More  generally,  when  //<"' 
requires  a  correction  according  to  the  binomial  coefficients  of  the  »/;"' 
order,  an  error  exists  in  ./'"-"".  These  remarks  inipiy  the  necessity  of 
some  caution  on  the  part  of  the  beginner. 

It  will  be  observed  that  when  either  the  first  oi'  last  function  of  a  series 
is  in  error,  only  the  first  or  the  last  term  in  each  order  of  difference  will 
be  affected,  and  only  by  an  amount  numerically  equal  to  the  erroi-.  Hence, 
in  such  cases,  the  method  above  explained  is  of  little  value. 

In  general,  it  may  be  stated  that  when  errors  have  been  dis- 
covered by  differencing,  it  is  advisable  to  re-comjiute  the  values  in 
question,  when  the  data  for  the  calculation  are  available. 

General  Properties  of  Differences. 

11.     Let     F{i),  F{t-\-w),  F{t^2,oj), represent  any  series 

of  tabular  functions,  whose  diff'erences  are  taken  as  in  the  schedule 
below  : 


Function, 
F(T) 

J' 

J" 

J'" 

Jfn) 

J<n+1) 

F{t) 

-v 

F{t  +  m) 

A' 

■h" 

JJ" 

F{t  +  2„>) 

zlj 

v 

..7/" 

/''('+ 3<u) 

. 

J," 

J." 

4',"' 
j(„) 

/  (.1+11 
0 

.   .   . 

Flt+Sm'] 

JJ 

F[t+{s  +  \)w'] 

-1' 

-tj' 

/],'" 

F\_t+{s  +  2)^,'] 

-J".+. 

/I'll 

A'' 
At. 

J(,.+l. 

Alt' 

1()  THE  THEORY  AND  PRACTICE  OF  IXTERPOLATION. 

AVe  shall  aswnnu'  tliat  F (T)  is  a  (iiiitc  and  (•(mtiniious  function, 
and  tliat  /''(/-j-sw)  is  capablr  oi'  f\])aiisi()n  in  a  series  of  ])owers  of 
SO),  within  the  limits  of  the  given  taljle;  then,  denoting  the  suecessive 
derivatives  of  F{T)  by  F\T),  F"{T),  etc.,  we  have,  by  TA^^.OR's 
Theorem,  the  following  expressions  : 

Fit)  =  F(t) 

Fif  +  o>)  =  /XO  +    <"/'"(0  +        ^  F"  (t)  +       '^  F'"  (0  +         I  F^''  (0  + 

F(f  +  2.>)  =  F(f)  +  -2<oF'(f)+    4 -/'-'(/)+    8 1' /'""(/)+    16'^r-(/)+.    .    \     ^^^ 

i^(^+;;<„)  =  /''(/) +  ;!a,/"(<)+    qI^F" (f)  +  27 '^F"> (t)  +   8l^i^-(0  + 

/'(^  +  4a>)  =  Fit)  +  io>F(f)  +  16'^F"{f)  +  CA'^F"'(t)+256^F'^-{t)  + 

Differencing  these  valnes  of  the  functions  in  the  usual  manner, 
we  obtain  successively  the  expressions  for     J',  J",  .<!'"  .  .  .  .,     as  follows  : 

j;  =   ^F'it)+       I   F"{t)  +        ^'   F'"{t)  +  ^'  /"v(i)+  .    .    . 

OJ*" 


4'  =   o,/"(0+  3^'   F«{t)+    7    I   /'""(0+    15  ^  i^'^(0  + 


4/  =   ^F'(t)+   7^   /-'(0+  37  I    i?""(0+  175  ^'  i^'v(^)+   .... 


z/„'  -   «/'^'(0+    5  Ji   i''"(/)+  19  ^    F"'(0+    <>5  -^  F'Ut)+  . 


■•'  ..    co^ 


J/'   =   o,=  2^"(0  +  2a.'i?""(0  +  f§<u^i*''^-(0+    ....  (      .o) 

J„"   =    c«-i^"(<)  +3(0' i^ '"(<)  + iH^'«''?^'''(0+     ■     •     •     •  (        " 


//„"'   =   u,^F"'{t)+  iw^F'^f)  +   .... 

//,'"   =    o>^F"'(t)+  Rw'  F'-'-it)  +    ....  J     (3) 

It  will  be  observed  that  all  terms  of  the  expansions  (0)  are  of 
the  general  form,  K(x/'F'^^\t)  ;  where  K  denotes  a  numerical  factor, 
and  r  an  integer  which  increases  by  unity  as  we  proceed  from  any 
term   to  the    next   term  following.     Hence,  the  diferences  will   contain 


THE    TIIKORY    AND    PRACTICE    OF    INTERPOLATION.  17 

only  terms    of  tliis  form.     We   thus    see,  n    priori,  that    aii}^  difference 
of  the  //"'  order  must  he  of  tlic   form 

Let    us    now  as.'^ione    what    appeal's   fi'om   (1),   (2),  and   (.'3)   to    he 
the  general  law;  that  is 

A   =    1  r   =   n 


leaving-  the  coefficients     B,  C,  D, 
We  therefore  assume 


undetermined  for  the  present. 


Ji"'    =  a,"F'"'  (t)  +  B«."+'i^' "+"(;)  +  Cw"+-F"'-^"  (t)  +  Du,"+'F'"+'"  (t)  + 


(4) 


Since  the  value  of  f  is  arbitrary,  we  may  write  f -\- co  for  t;  by 
making  this  substitution  iji  the  right-hand  member  of  (4),  we  evidently 
get  the  expression  for  the  »"'  difference  inunediately  following  Ji"', 
—  that  is,  the  value  of     Ji'j^^.     Hence  we  have 


.^i;\ 


„"F<"Ut  +  ,o)+Bo,"+'F^"+''(f  +  m)+Cw"+-F"'+-'(f  +  w)  +  I)m"+'F"^--'\f +  „,)  +  .    . 


Developing  the  functions  of  the    right-hand  member  by  Taylor's 
Theorem,  we  find 


fo)     - 

•+1  " 

^    w" 

^.,,„ 

7)  +  a,i^'"+''(0    +~F 

"-(0+1 

F"'+^l  {f)+  .  . 

■  ■ 

+  Bo,"+' 

F'"+"(f)  +  o,F '"+"-' (f)+   '^V'"+'"(<)+    .... 

-1 

+  0,,,"+'- 

F'-'+-\t)  +  ,.,F'"+'' (t)  +    .    .    .    . 

+   Z>(0"+' 

+  .  .  .  . 

Collecting  the  coefficients  of    F'^^f^),  F"'+"{t),  .  .   .   .,     we  obtain 


//(;>,   =   a,"  F<"'  (t)  +  (B  +  l)u>"+'F"'+'\f)+[  C  +/i  +-  )o,"+-i^'"+2'  (f) 


(5) 


+  (  />+ C+ I +1  )  a,"+'i^'"+^' (0  + 


Siibtracting  (4)  from  (5),  and  observing  that      /i:;.\  — ./i"'  =  ./i"+",     we 
get 

j(n+i)  _   ^"+^F>"+'\t}+fB  +^\  oi"+^F<"+'''{f)  +  f  <:'-{-?-+ ^^  o,"+'F^''+''(f) 

+  (^+1+1+1)-""  ^"■"'(')+  •  •  •  • 


^^^ 


18  THK    THEORY    AND   PRACTICE    OF    INTERPOLATION. 

If,  tlu'i'ofoi'i'.   we  ])iit 

li'  =  ^+^ 

^'=^+1+1  )  (C) 

-  =  -444 


we  have 

J^;''^^'^  u>''*^F^"+'\t)  +  B'm''+-F'"^-'{t:)+C',.>"+^F'"^''\f)  + D'm"^F^''*''{f)+  .    ...       (7) 

Hence,  if  the  general  form  of  expression  assumed  in  (4)  is  true 
for  the  index  //,  it  follows  from  (7)  that  it  is  also  true  for  y/-|-l; 
but  we  see  by  equations  (1),  (2),  and  (3),  that  the  law  obtains  for 
V  ^1,  2,  3,  respectivel}' ;  hence  it  holds  for  n  =  4;  and  so  on 
indefinitely.  The  expression  (4)  is  therefore  true  for  all  positive  inte- 
gral values  of  n. 

12.  AVe  have  now  to  detenuini'  the  coefficients  B,  C,  J),  ....  , 
of  equation  (4).  These  quantities  are  evidently  functions  of  n  and  s, 
and  will  be  determined  in  the  following  manner  : 

First,  we  take  .-!  =  0.  and  determine  the  constants  for  ./y',  which 
we  shall  denote  for  this  purpose  by     B„,  C„,  D„,  .... 

These  values  are  found  by  induction,  thus:  the  relations   (H)   give 

B„+^,   Cn+x,    A+i ill   t^'i'i"*^   f>l"     J^n,   C„,  n,,,  .  .  .  .      Making 

n  =  l,  we  take  By,  Ci,  D,,  .  .  .  .  directly  from  the  first  of  the 
equations  (1);  a  continued  apphcation  of  ((>)  therefore  gives  succes- 
sively the  values  of  B.,,  B-^,  B^,  .  .  .  .  -S„_, ,  B„.  Similarly,  we 
derive  C'„,  //„,  ....  Hence,  the  coefficients  of  (4)  become  known 
for     s  =  0. 

Second,  the  coefficients  of  /;"'  easily  follow  from  those  of  j;,"'; 
for  it  is  cleai-  from  the  schedule  of  ^^11  that  //j"'  is  related  to 
F{t-\-S(o)  in  precisely  the  manner  that  /),"'  is  related  to  F{t)- 
Hence,  if  Ibr  l)revity  we  write 

we  shall  have,  since  the  value  of  /  is  arbitrary, 


THT5    THEORY   AND    PRACTICE   OF    INTERPOLATION.  1!) 

Then,  expanding  ^(/-f- s"w)  in  i'  series  of  powers  of  sw,  we  arrive 
at  an  expression  of  the  form  (-4),  in  whicli  the  coefficients  are  fnlly 
determined  functions  of  71  and  s. 

To    perform    the    steps    indicated,    we    take    from    tiie    first    of  tlie 
equations  (1)  the  following  values: 

^1  =  i  C\  =  i  I\  =  ,V  ■    ■    •    •  (8) 

To  find  B„:     By  repeated  application  of  the  first  of  (6),  we  have 

B,  =   5,  +  i 
B,  =   B,+\ 


Hence,  by  the  addition  of  these     n  —  1     equations,  we  get 

7?„  =   7?,  +  *(//-l)   =  i  +  i(M-l) 
To  find   C„:     Using  the  second  of  ((j),  we  obtain 

c,  =  c,  +  i  /;,  +  -■ 


n 


('..  =    (',..-,  +  i  B„_,  +  I 
whence,  by  addition,  we  find 

C„   =    C,  +  i(5,  +  /i„+  .    .    .    .  +5„_,)  +  1  (w_l) 
Since     Ci  =  i,     this  gives 

r=ii— 1 

C„  =  ^{B,  +  B,  +  ....+  /?„^0  +1=  \^B,+  l 

r=  1 

But,  from  (9),  we  have     B^  =z  ',  ;     hence  we  get 


c..i2,.  +  ?  =  t 


+  S-24»»  +  l) 


To  find  Z>„:     Again,  from  ((>),  we  derive 

D,   =   A  +  i  C  +  ,\  B.,  +  ,->, 


(9) 


(10) 


^,.   =   iA.-i  +  i  C,._,  +  1  2}„_i  +  jV 


20 


THE    THEOKV    AM)    rKACTlCJh;    OF    IKTEUP0LAT10^•. 


whence 


r=l  r= 1  r= 1 

From  (10),  we  have 


24 


I) 


=  .v2;''^+-S''+S 


or 


=  ,v 

''Ijn-l)i2n- 

-1)" 

+  A 

n(«  — 1)"]  ^   n'-' 
2              24 

^;„  = 

£("+!) 

( 

ai) 


In  like  manner,  the  process  might  be  extended  to  the  vakies  of 
En,  F„,  .  .  .  .  ;  but  the  results  already  obtained  are  here  sufficient. 
Substituting-  in  equation  (4)  the  values  of  B^,  C,,,  and  />„,  given  by 
(9),   (10),   (11).   (remembering  that    these    values    suppose   .s  r=  0),  we 

have 

(12) 

Ji"'  =  o>"F-"  (t)  +  "■  cu"+>F'"+>'  (0  +  ^(3"  +  1)  «;"«/'>"+='  (0  +  ^  («  +  l)  o."+^'7^'"+»'(0+  .    . 

We  now  obtain  from  (12)  the  expression  for  ./i"\  As  already 
pi'oposed,  we  write 

Then,  as  shown  above,  we  shall  have 

,2    2  ,a^8 

Jl")     =     ^(t  +  S,n)     =     *  CO    +  .50,   *' (0    +  ^ *"(0    +   ^—   *'"(')+       .... 

[2  (|i_ 


=   U''F'''' (t)  +  Jiy+'F'"+'' (f)+  C,,,o"+-F'"+-'  (t)  +  J),,o,"+-'F^"^'''  (t)  + 

+  SW  (o,"F-+''{f)  +  n„io"+'F'"+--{t)+c,y*U''"'+^'(f)+  .  .   .  . 


+  LZ.   oy"F^"+-'  (t)  +  7.'„<„"+i F'"+»'  (0  + 

[2     V 

+  f^'(..^-(.)+.    .    .    .)     +.    .    .    . 

Upon  arranging  this  expression  according  to  ascending  powers  of 
o),  we  get 

j(...  =  ^n;^,.„  ^f^  +  (y,'„+.s-)  ..."+'7^' ■■+>'(/■)+  (c'„+  n„s  +  ^)  ^•'+-F'"^"-'(f)  (13) 


Hence,    substitnting    the  foregoing  values    of  B„,   C„,  and  B„,  we 


DR.  C  ;--.  McEWEN 

THE    THEOHY   AND    PRACTICE    OF    INTEKPOLATION.  21 


find     tliat     the    values    of     i?,  (\  I) 

I'ollows: 


ill    e([ii:iti()ii    (I)    are    as 


I>  = 


L'4 

n  +  'ls 


(14) 


'n{ii  -f- 1  ) 


+  ■-•(  "  +  ■-•) 


These  results  are  easily  verified  by  substituting  special  values  of 
n  and  s,  and  comparing  Avith  the  coefficients  in  equations  (1),  (2),  (H); 
thus,  putting  s  =  1,  and  taking  ?i  =:  1,2,  3,  successively,  we  ob- 
tain the  numerical  coefficients  in  the  expansions  of  J,',  ./",,  and  /,"', 
i-espeetively. 

13.     lieviarlablc  Formal  Relation  hetween  the  Expressions  for    ./;"' 

and  Jj. —  The  coefficients    7?„,  C„,  D„ ,    in  the  expression  for    /„", 

may  also  be  determined  by  the  following  method,  which  not  only  is 
shortei-  than  the  above,  but  also  possesses  the  advantage  of  showing 
a  direct  relation  between  the  expressions  for  j;"  and  z/„',  respectively. 
Retaining  the  above  notation,  we  write   (12)   in  the  form 


//„'■"  =  u,"r""{i)  +  «„  0,"+' /■'"•+"  (i!)  +  c„<u"+-F'"+-'(<)  + 
We  now  let 


(15) 


(15«) 


be  an  auxiliary  expression,  such  that  the  coefficient  of  >/"+''  is  the 
coefficient  of  aj"+'-i^'"+'' (0  iu(lo).  Writing  n  +  1  for  n  in(13aj, 
and  using  the  relations   (6),  we  have 


<^,.+i(2/)   =   ,'/■'+'  +  (  J^.  +  ir  ]  t/"^-'  +  {  (-■„  +  ^"  + 


11 


(2_  13 


!/ 


(16) 


^(^'"  +  ^'-^f4j^"^^^-    ■    • 


Again,  since  the  coefficients  of     q,  (//)      are  those  of    j^',     we  ob- 
tain from   (1), 

'■'  Jt  J 

/.'*  /y*  //' 

(17) 


'h  {!/) 


i;-         !/■'        ip 


11      E      li 


22  THE    THEORY   AXD    PRACTICE    OF    IXTERI'OI.ATION. 

By  re-ai-ranging  the  terms  of  (16),  we   iiiul 
?/"+-     //"+■'     »"+■' 

\  E        li         li 

//"+<        y"+^        */"+'■' 


/                        ,^..+5           ,,«+«            ,/n+7 
+ 


+ 


=  (.'/  +  -!^+f +f  +  •  •  •  •  )(^"+^"y"^'+^".'/"^'+^'..y"^'+ •  • 


11      li      li 
Hence,  by   (15«)   and   (17),  avc  have 

'r„+i  =  'fi  •  •(„ 
Taking     /^  =  1,  2,  3,  ....  w —  1,     successively,  we  iind 

r/.i  =  <r'ig-.-  

gr,     =      'iPl<if4  7..  =      VlVn-l 

Multiplying    these    equations    together    member    foi-    member,    and 
cancelling  the  common  f^ictors,  we  obtain 

<ip„  =  (.qp.)"  (18) 

Therefore,  by   (17),  we  have 

■■■cp„iy)  =  .r+^r+'  +  2j(3"+i)//"^Hjg(«+i)r+'+  ■  •  ■  •         <i9) 


THE  THEOKY  AM)  PRACTICE  OF  INTERPOLATION.  23 

Conipai-ino-  coefficients  in   (15</)   miuI    (10).  \vc  find 

_  w4  4  o 

Suhstitntiny  these   valiu's  in    (15),  the   hitter  becomes  . 

which  agrees  with   (12). 

These  results  may  be  conveniently  expressed  symbolically:  thus, 
let  us  represent  the  quantities  J^,',  J/',  ./„'",  .  .  ../„""  by  ./„,  J„-=,  J„%  .  .  ./;;; 
and  tor  a,F'(t),  w'F"(f),  w'F"'(t),  ....  (o"F""{f)  let  us  write  the 
symbols      I),  7)\  />',  ....    I)",     respectively;    then  we  shall   ha\e 

//-    n^    i>*     !)''■ 

/  ,       I)'      U'      D' 

=   ir  +  '^:  7r+'  +  A , ;{,,  + 1 ,  D"+"-  +  ^  ( «  + 1 )  l>"^'  + 

14.  Theorem  Y. —  The  n"'  differences  of  amj  rationed  iideiiral 
expression  of  the  «"'  degree  are  constant.  If  the  f/eneral  form  of  the 
function  is  F(T)  =  aT"  +  /8T"-'  +  yT"-'-+  .  .  .  .  ,  the  constant 
value  of    ./""     is  w"('.  [n . 


For,  from  the  nature  of  the  function,  we  have,  evidently, 

and  F"'+"(t}  =  F'''+"-'{t)  =    ....    =0 

Hence,  from  (4),  we  have 

J',"'   =  io"  F^"'(f)   =   o)"«[»  (22) 

The  theorem  is  therefoi'e  true,  whatever  the  value  of  the  constant 
interval   a..     Several   examples   have    already  occurred:    in  §2  we  have 


24  THE    THEORY    AXD    PRACTICE    OF   INTERPOLATION. 

the  differences  of    F (T)  =  T'—10T'—20;    here    ?i  =  4,  a  =  1,  0;  =  1. 
Hence,  by  (22),  we  get 

—  the  vahie.  ah-eady  found  by  differencing. 

In  Example  I  of  §9,  F (T)  ~  T\  m  =  1;  we  there  obtained 
for  the  value  of  the'  third  ditference 

J'"  =  (i 

which  agrees  with  the  theorem. 

Again,  in  Example  I  of  §10,  F{T)  =  2T'—25T—iO,  w  —  l; 
whence  the  theorem  requii-es 

J'"  =   „[3_    =  21i    =   12 

which  is  the  I'esult  already  obtained. 

15.  Theorem  \1. —  //'  the  n"'  differences  of  a  series  of  quantities 
(tabulated  for  equidistant  values  of  T )  are  constant,  the  given  quanti- 
ties are  the  tabular  values  of  a  rational  integral  function  of  the  form 

F{T)=aT"^liT"-'^yT"-'-\-   .... 

This  proposition  is  the  converse  of  Theorem  V,  and  is  pi'oved  as 
follows: 

Let  F(T)  denote  the  function  whose  true  matheinatical  values, 
tabulated  for  the  given  values  of  T,  form  the  given  series  of  quantities. 
From  (4)  and  (5),  we  see  that  the  expressions  for  z/^"'  and  .j;;'i  agree 
only  in  their  first  terra,  ai"F'"'{t);  the  remaining  terms  of  like  order 
in  u)  having  unlike  coefficients.  Hence,  the  conditions  necessary  in 
order  that     J""'     shall  be  constant  throughout  are  as  follows: 

First,      that     a)"F"'\t)     does  not  vanish; 

Second,  that     w"+'F'"+"{t)  =  w"+'F"'+-\t)  =  ....  =  0; 

But,  since  w  cannot  vanish,  these  conditions  reduce  to  the  form  — 


^""  (*)i'  I  (23) 


If  now  we  put 

7'  =   f  +  T  (24) 

then,  by  Taylor's  Theorem,  we  have 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION.  25 


r"  ■^"+' 


By   (23),  this  gives 

F{T)   =    F(t)  +  TF'(t)  -\-  .    .    .    .+ /.'•■-ii(o+^    /''""(C)  (25) 

In— 1  In 

in  whicli,  \vc  obsi-rve,  tlie  cofltifieiit  of  t"  caiiiiot  vanish.  Substituting" 
in    (25)   the  value  of  r  given  hy    (24),   we  obtain 

F"  it)                                      F""i.f) 
F(T)   =    F(t)  +  {T~t]F'(t)+  (T-t)' ^+.    .    .    .  +  (7-/)" ^ 

Sinee  t  has  a  fixed  value,  the  right-hand  menibei'  of  this  equation 
is  an  expression  of  the  7i"'  degree  in  the  variable  1\  and  hence  may 
be  wi'itten   in  the  form 

F(T)^aT"  + liT-'  +  yT"--+    ....  (•_'(;) 

Avhieh  establishes  the  theorem. 

16.  Convergence  of  the  Differences  in  Practice. — In  the  discussion 
of  Theorems  \  and  VI,  we  were  concerned  with  the  true  mathematical 
values  of  the  quantities  involved.  In  practice,  however,  the  absolute 
or  true  mathematical  values  of  functions  are  seldom  employed;  fre- 
quently, as  previously  noted,  a  function  is  tabulated  only  to  a  certain 
degree  of  approximation,  enough  decimals  being  retained  to  give  the 
desired  accuracy.  We  observe  that  in  such  cases  there  is  a  tendency 
of  the  differences  to  decrease  numerically,  and  usually  to  vanish  sensibly, 
as  the  oi'der  of  difference  progresses.  The  explanation  of  this  tendency 
follows    readily  from    equation    (4),   thus:    for   any  given    function,  the 

derivatives     i^""(0,  F"'+'\t),  F'"^-' (f) have  definite  values; 

hence,  the  value  of  w  may  be  chosen  sufficiently  small  to  render  all 
the  terms  in  the  second  member  of  (4)  insensible,  excejjt  the  first. 
When  this  condition  obtains,  the  value  of  J*"'  is  sensibly  constant, 
and  equal  to  oj"F""(t).  The  differences  of  F{T)  are  thus  practi- 
cally brought  to  a  termination  at  the  /t"'  order,  whether  the  function 
is  algebraic  or  transcendental. 

In  many  cases  the  values  of  the  successive  derivatives  converge 
rapidly;  the  chosen  value  of  w  may  then  be  quite  large,  and  yet  allow 
the  differences  to  sensibly  vanish  at  an  early  order.     This  is  equivalent 


26 


THE  THEORV  A^■D  PRACTICE  OF  INTERPOLATION. 


to  the  obvious  statement  that,  when  a  function  is  to  be  tabuhited  so 
as  to  ditterence  readily,  the  interval  of  the  argument  ujust  be  decided 
by  the  manner  in  which  the  given  function   varies. 

To    exemplify    these    principles,    we    take    the    following    table    of 
seven-figure  logarithms : 


T 

Log  T 

J' 

J" 

J"' 

1.00 
1.01 
1.02 
1.03 
1.04 
1.05 
1.06 

0.0000000 
.0043214 
.0086002 
.0128372 
.0170333 
.0211893 

0.0253059 

+  43214 
42788 
42370 
41961 
41560 

+  41166 

-426 
418 
409 
401 

-394 

+  8 
9 

8 

+  7 

In  this  case,  w  =  0.01,  t  =  1.00,  ^  +  w  =  1.01,  t-\-2(o  —  1.02,  etc. 
To  serve  our  present  jiurpose,  we  here  transcribe  from  (1),  (2).  and 
(3),  the  following  expressions: 

^o'   =  -^'"(0  +  'J^"W+^'^""(«J +^^'"^0+  •   ■   •   • 

j;'  =  „/^i*"'(o +  a,'i^"'((;)  +  T-vu)^i'"^(0+  ....  [  (27) 

j;"  =  <,/7^""(^)+|co*i?"^(0+  ....  \ 

^;v   _   „4;riv,Y)+    ....  / 

Since     F{T)  —  log  T,     we  have 

V<{t]  =  +  -Vr'  ,   F"(t)  =  -Mt-"  ,   F"'{t)  =  +2.)^-'  ,  F'--(t)  =  -MIt-'  ,  .    .    .    . 

where  Jf  is  the  modulus  of  the  connnon  system  of  logarithms,  =  0.434294. 
Hence,  wnth     t  =  1     and    w  =  0.01,    we  find 


^F'  {t)    =   +0.0043429,4 
o'^F"(f}   =  -0.0000434,;! 


u>^F"\t)    =    +0.0000008,7 

„r/r'v  (t)  =  -0.0000000,3 


Substituting  these    numerical    values    in   (27),  we    obtain,  in    units 
of  the  7th  decimal, 

j;   =    +43214  /„"   =    -426  J/"   =    +8  J^'^'   =   0 

which    agree    substantially  with    the    results    obtained    above    by   direct 
differencing.     The    rapid    convergence   of  the  differences   is  thus  seen 


THE    THEORY    AND    PRACTICE    OF   INTERPOLATION. 


27 


to  be  due  to  the  small  value  of  the  interval  w,  which  makes  the  term 
w'F"'{f)  appreciable,  but  renders  aj'F^'(f),  a/'F^it),  ....  (|uite 
insensible;  accordingly.  /'"  is  the  last  difterence  which  we  need  take 
into  account,  the  remaining  diffei'ences  being  practically  zero. 

We  may  add  that  if  the  values  of  T  in  the  jjresent  table  were 
100,  10],  etc.,  instead  of  the  givi-n  values,  the  intei'val  o)  would  become 
1  instead  of  0.01,  and  hence  oi,  or,  «/',  <</,  ....  would  not  converge 
as  above.  We  should  then,  however,  have  /  ^  100  instead  of  1, 
which  would  cause  the  successive  derivatives  to  converge  i-apidly,  as 
is  obvious  from  the  general  expression 


F""(t)   =   (-1)"-' .¥[»-..  - 
Furthermore,  the  differences  of    F(T)     contain  only  tei-ms  of  the 


form 


whei'e  K  denotes  a  numerical  factor;  hence,  since  the  values  of  w  and 
t  are  both  increased  one  hundred-fold  by  the  assumed  change,  it  is 
evident  that  the  general  term  Kco"F^"\t)  is  not  altered  thei'eby. 
The  differences  are  therefore  unaltered  by  the  proposed  change;  this 
conclusion  is  confirmed  by  the  consideration  that  the  assumed  altei'a- 
tion  in  T  would  merely  change  the  logarithmic  characteristic  trom  0 
to  2,  and  thus  would  not  affect  the  resulting  diffei'ences.  These  ob- 
servations illustrate  the  case  of  a  tabular  function  whose  successive 
derivatives  converge  rapidl}',  whereby  a  comparatively  large  argument 
interval  may  be  used,  and  yet  allow  the  resulting  series  of  differences 
to  converge  as  i-apidly  as  may  be  required. 

17.     As    a    second    example,    we    consider    the    following   table    of 
cubes : 


5.16 
5.21 
5.26 
5.31 
5.36 
5.41 
5.46 


ys 


J" 


J'" 


137.39 
141.42 
145.53 
149.72 
153.99 
158.34 
162.77 


+  4.03 
4.11 
4.19 
4.27 
4.35 

+4.43 


+  0.08 
.08 
.08 
.08 

+  0.08 


28  THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 

We  have  alirady  seen  ( TlieoreiirV)  tliat  when  the  true  mathe- 
matical values  of  I'''  aiv  tabulated,  the  third  diffei-ences  are  constant, 
the  fourth  differences  being  the  first  order  to  vanish.  In  the  present 
table,  however,  only  two  decimals  have  been  retained  in  J'^  whereas 
the  true  value  involves  six  places.  To  this  degree  of  approximation,  the 
thii-d  differences  are  entirely  insensible;  this  follows  from  Theorem  V, 
which  gives  for  the  constant  value  of    /'"  — 

J"'     =     <u»«  [3 

In  this  example  Ave  have 

u,  =   0.05  u  =   1 

and  hence 

J'"  =   (0.05)"  X'i   =  0.00,075 

which  is  insensible  when  only  two  decimals  are  concerned.  Thus,  in 
the  approximations  so  frequently  used  in  ]>ractice,  the  differences 
generally  terminate  (either  absolutely  or  approximately)  at  some  ordei* 
eai'lier  than  would  occur  if  the  true  mathematical  values  of  the  function 
Avere  emploj'cd. 

It  may  be  added  that  the  above  example  affords  an  illustration  of 
Theorem  VI.     For,    since    the    second    differences    are    here   absolutely 
constant,  it  follows    from    this    theorem  that  the    tabular  quantities  are 
the  true  mathematical  values  (corresponding  to  the  given  values  of  T) 
of  some  function  of  the  form 

F{T)  E  uT-  +  (iT+  y 

Thus,  in  i)articular,  if  the  student  tabulates  the  function 

F{T)  E  16  (T''- 5.3325  2'+ 9.476975) 

foi-  T  ^  5.16,  5.21,  ....  5.4(5,  and  retains  all  decimals  involved, 
he  will  find  his  tabular  numbers  identical  with  the  above  series. 

18.  To  Exjiress  co''F^"\t)  in  Terms  of  J\r,  4',"+'',  Ji"+",  etc. — 
The  j)roblem  consists  in  reversing  the  series  (15),  w^hich  expresses 
z/i-     in  terms  of     o,"F"''{t),    <«"+'F"'+"(/),    .... 

Let  us  denote  <-/^^'^'(/)  by  Xr\  then,  Avriting  successively, 
n,  n  +  l,  n-\-2,  ....     for  u  in   (15),  we  have 


THE    THEORY   AND   PRACTICE    OF    INTERPOLATION.  29 


from  which  we  obtain,  by  transposition, 

a-,,      =    Ji"' -/>'„.'•„« -C,,r„^,- A,  .T„^,-  .     .     .    .     \ 

^).+l   ="     ^o"  -'^  .1+1  •'•■.1+2  ^n+l^n+3  -^.i+l*^;.+4  .       .       .       .       f 

a;„+o=  4V'+-'-^„+...T„+3  -  C„+„.x„+< -Z)„+„.-B„+.  -     .    .    .    .     /  "  -' 

The  second  of  the  equations  (29)  g-ives^  a  vah\e  of  «„_(.,,  which, 
substituted  in  the  tirst  equation,  gives  a?„  in  terms  of  j,v",  J{,"+",  j„^„,  .i„+,,  . . . ; 
substituting  in  tlu'  latter  expression  the  vahie  of  x„^2  given  hj  tlie  tliird 
of  (29),  we  find  «„  in  terms  of  A*,"',  Jy^"",  z/i""^"',  .r,,^.,,  a-,,^.,,  ....  Con- 
tinuing this  process  of  elimination  indefinitely,  we  ari-ive  at  an  expi-es- 
sion  of  tlie  form 

-  The  coefficients     b,, ,  c„ ,  d„,  ....      must  now  be  determined.     From 
(15a)  we  obtain  the  following-  group  of  equations  : 

<f„+i=  y"+'  +  ^'„+,?/"+^+c^„+,i/"+=+A,+.//"+^+ .  .   .  .   f 


:) 


Comparing-  (28)  and  (31),  we  observe  that  tiie  latter  group  may 
be  obtained  from  the  former  by  writing  if,,  and  ?/''  for  /j;'  and  x^, 
respectively;  the  algebraic  i-elations  in  both  groups  are  otherwise  identi- 
cal. Hence,  if  from  (31)  we  seek  to  express  //"  in  terms  of 
<Tn}  <fn+i>  fn+2  J  •  •  •  -,  the  proccss  of  reversion  will  be  identical  with 
that  which  gives  ,r„  in  terms  of     ./j"',  J,V+",  .    .    .   .  ;     hence  Ave  must  find 

!/"  =   'I..+ ''..ff,.+i  + '',.<r..+2  +  <^l.''l..+^+  ■    ■    ■    ■  (32) 

the  coefficients  being  those  of  (30).     Therefore,  by   (18),  we  have 

y"  =  ,,;■  + /,„f/;'+i +c„,r;«+,/,T,"+^-'+  .    .    .    .  (33) 

Taking     n  =  1,     in   (30)   and   (33),  we  obtain 

x,  =  4/  +  '\ 4,"  +  '•i -J^'"  +  (k.1,r  +  .   .    .    .  (34) 

and 

.'/  =  gi  +  /'i'fi^  +  '-i<h'+'^T'.^+  •    •    ■    .  (35) 


DR.  GEO(  Ei^ 

30  THK    THEOin     AND    l']{ ACTIfl",    OF    INTERPOI^ATION. 

Fi-diu    (17).   l)v   adding-  unity   to  cacli   iiicnilxT.   wc  liavo 

1+q,   =    -i+!/+^^+'^+(+    .    .    .    .  ="'  (36) 

ii       L       li 

or 

//   =   log,  ( 1  +  (/ 1 )  ( 37 ) 

■•■  !/  =  'h-^yr  +  :i'h"-i'r.^+  •  •   •  •  (38) 

Conij>aring'  coefficients  in    (3o)   and   ('^8),  we  find 

h,    =     -i  '•,    =     +    :\  <h    =     -i  ....  (39) 

Substituting  these  values  in   (34),  we  obtain 

,,.    =  ^F'(f)    =   j;_^+-| -_^_+    ....  (40) 

Again,  from   (38),  we  derive 

■"■■  =  ('^'-T+T-T+  ••■•)"  ^''^ 

••■  .'/■'   =   gr-g--^"^'  +^^""  +  '"'>'fi^'-  3§<"  +  2)(n  +  3)g.r»+   ....  (42) 

Equating  coefficients  in    (33)   and   (42),  we  find 

K   =    -|    '     '•..   =  +y4(3"  +  ">)     ,    d„  =    _^(«,  +  2)(«,  +  3)     ,       ....  (43) 

These  values  being  substituted  in   (30),  the  lattei-  becomes 

x„E  o>"F'"'(f)   =  4V''-^./V'+''+ J4(3"+-"^). '/;,"+=' -^(w  +  ii)(«  +  3)4V'+''+  .    .    (44) 

Using  the  symbolic  uotation  adopted  in  (21),  we  have  the  follow- 
ing expressions : 

D  =  J.-iJ^+lJ^-iJ^  +  lJ^-  .    .    . 

(45) 

i)"=  (.4-i.^„^+A-'o^-iC+  ■•■)■■ 

=  4^-|4';+i  +  ^(3«,  +  r,)../''+^^-^(,,  +  2)(«+3). /;■+»+  .    . 

19.     Effect   of  a    Change   in   the   Argument   Interval   m,   upon  the 
Magnitude  of  the  Several  Orders  of  Differences.  —  Let  us  now  suppose 


THF    TilKOKY    AND    I'KAf'TICE    OF    INTERPOLATION. 


31 


that  a  second  tabulation  of  F(T)  has  hciMi  inado,  differing  from  the 
first  only  in  the  value  of  the  interval,  to.  Let  o>  =  uuo  be  the  in- 
terval of  the  argument  in  the  second  table;  denoting  the  differences 
by     8',  8",  8'",  .  .  .  .,     the  new  table  will  run  as  follows: 


T 

F(T) 

8' 

8" 

8'" 

8iv 

.... 

t 

F(t) 

So' 
8r,' 

i  +    III  01 

F(t+   1,10,) 

K" 

8„"' 
8'" 
8„"' 

t  +  2»/(u 

F(t  +  2iiiu,) 

8," 

So"- 

t  +  3w/o) 

F(f+3ino,) 

8.," 

8i'^ 

f  +  4'/H(i) 

F(t  +  ivio,) 

We  proceed  to  investigate  the  relations  between  8',  8",  8 ",  .... 
and  //'.  J",  z/'",  ....  No  restriction  is  placed  upon  the  value  of  m; 
in  the  applications  of  the  resulting  formulae,  however,  in  will  usually  be 
regarded  as  a  positive  proper  fraction.  The  second  tabulation  will 
then  give  the  function  for  closer  values  of  T  than  the  first. 

Since  the  value  of  o)  is  arbitrary,  we  may  write  niM  for  to  in  the 
right-hand  member  of  (lo),  and  thus  obtain  the  expression  for  8!,"'; 
making  this  substitution,  we  find 

8,V"    =    iii\,"F f)+ n„m"+'o,"+''F'"+'Uf)+  C„m"+"-o,"+-'F'"+"{f)+  .    .    .    .        (46) 

If,  as  above,  we  write  ,;■,.  for     «)''F"'\t),     this  equation  becomes 

8J,'"    =    »'".-/•„+  £,.»("+»  a-„+i  +  C„m"+^r„^.^  +   ....  (47) 

From  (30)   we  obtain,  in  siiccession. 


(48) 


Eliminating     .i„?  ■^■«+ij  •   •  •  •     from   (47),   by  means  of  (48),  there 
results  an  equation  of  the  form 

8-'  =   ///",/r'  +  /3„.J,V+>'  +  y„./r-^'+  .    .    .    .  (49) 

which,  for     «  =r  1,     becomes 

8„'  =  .«j,/  +  A.y„"  +  y..V"+  ....  (50) 


32  THE    THEORY    AM)    PRACTICE   OF    IXTERPOLATION. 

IS^ow  let 

«„  =  w" //"+/?„'"""*"'//"+'+  C'„  »'""•"-.'/"""+  .    .    .    .  (51) 

be    ail    auxiliary  expression,  sucli    that    tlie    coefficient    of    y"+''     is    tlie 
coefficient  of    .»■„_,_,.     in   (47). 

From   (33)   Ave  obtain,  in  succession, 

.'/"    =  gi+ /'„  </!+'  + '■„<■/■;'+•- +  rf,.<ri'+"+  ....        1 


IS'ow,  to  eliminate     y",  .'/""'"' from  (51),  by  means  of  (52), 

we  must  perform  precisely  the  same  algebraic  steps    as    in    the  deriya- 
tion  of  equation    (49)    from   (47)   and   (48);     we   shall   therefore  obtain 

z„  =   .r,f;  +/3„(ri'+'  +  y„.r;+'+  •    ■    ■    .  (63; 

and,   for     //  =  1,     we  haye 

■M  =  ""/i  + /3i'r,'  + yi<(i'+  ....  (54) 

XoAV  the  equation   (51)   may  be  written 

z,^   =   ()»'//)"  + 7>„  (//(//)"+'+  f,'„( ■/«//) "+-+  .... 

Whence,  by  (15a),  we  have 

«„    =    (f„  (m.)/)  (55) 

and  hence,  also, 
or,  by   (17), 

12.        li        11 

,-.   1  +  Sj   =   e'"'  (56) 

Also,  from  (3(5),  we  have 

1  +  (f),  =  fl* 

the  combination  of  which  with    (5(5)   gives 

H     ,  .,    ,  .     ,  ,    w(ot  — 1)       .,  i)i(m  —  l)  .    .  (»i  — )•  +  !) 

[i  IT. 

or 

7/1  (m  —  1)     .  ml'/n  —  l)  .    .    .  tm  —  r  +  1) 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


'sn 


(58) 


Comparing   (54)   and   (~u),  we  liiid 

_   m(m  —  l)  m{iii  —  \){m  —  2) 

A  -  -^        '         V.  =  - 

Substituting  tiicsc  values   in   (oO),  we  obtain    tlic  following  fuiida 
mental  relation : 

tn(iii  —  1 )  . 


(V  ;  ^  /       m(iii  —  I)  _,  „ 


■  + 


r  + 


Again,  using  the  relation     (/„  =  (/;,      we  obtain  from   (55) 

■~,.  =  T.,  ('".'/)  =  ^Ti  ('".'/)  I"  =  -;' 

Hence,  from  (57),  we  find 

/  m (m  —  1 )      .      m(m  —  l)  (m  —2) 

2..  =  (""/i+-^; — -<ri  +— rr^ -(i'i'+  ■  ■  ■  ■ 


■III"  (jii  —  1)  fjf['+- 

m"  {m  —1)  (jf;'+'+   . 


(59) 
(GO) 


[2  "      ■  (3 

Expanding  and  factoring,  we  obtain 


K  =  "'"  f  i"  +  ^' '«"  ('"■  - 1)  <-f ;'+'  +  J 


(3»  +  l)»i-(3«  +  5) 


+ 


48 


»j(?t  +  l)  III--2  (w.-+3«  +  l)  /«  +(?i  +  2)(w  +  8) 


(Gl) 


Equating  coefficients   of  like  powers  of  r^j  in   (53)   and  (01),  we  have 


^n  =   2"^"('"-l)    '      V"  =  24 '""<^"'~-^) 


(3m  +  1)»(-(3«  +  5) 


Substituting  these  vahies  in  (49),  the  latter  becomes 


8i"'   =   w"  Ji">  +  ij  »i"  (?»  - 1)  J,V'+"  +  ^  «t"  (;;( - 1) 


(3M  +  l)«i-(3»  +  5) 


+  _„,"(„,  _1) 


i{n  +  l)m"--2(:n"-  +  Zn  +  l)m  +  {n-\-2){n  +  ^) 


....  (G2) 

Ji»+^>  (63) 

.-'ll  '        .       .        .       , 


Finally,  we  may  symbolize  these  results   by  the  following  expres- 
sions :  (64) 


8    -  viJ  I  "<"'-^)j2|   H^-m''"'-^)  ji^   M»^-l)--(^-3)j4  I   mim-l).  .{m-ij)^ 


11 


li 


li 


V  =  ( iiij, + ^^^^j^+  -(— y— -)^„3^ 


15. 


'o^+ 


K 


=  ».H=+  „r\m-l)J^'+'^(m-l)  (7..-ll)4,^+g(;»,-l)(».-2)(3w,-5)  J„s^. 
=  (;»J„+"^^"'~^^J/+ J=  ».''J/+  i-  m^(;«-l)j;  +  ^(».-l)(5,«-7)  J/+ 


8;  =  (»/ J„+  "'^"'^   ^^  J;^+ J=  /»U;+  2».^(;«-])  J/+  '^^  (m-1)  (13m -17)  j;+ 

o  =        /      .       iii.(m  —  l)    ^  ., 


=  in'J„'+  f  m\in-l)J„'+im'(m-l)(_4iH-5)JJ-\- 


34 


THE  THEORY  AND  PKACTICE  OF  INTERPOLATION. 


20.  Theorem  YII.  —  If  the  n"'  differences  of  a  given  series  of 
functions  are  tiumerically  large  as  comjyared  xvith  all  the  following 
differences,  then,  if  the  series  he  re-tahulated  ivith  the  argument  interval 
m  times  its  original  value,  the  n"'  differences  of  the  new  series  will 
he  apj)roximatelii  m"  times  the  corresponding  n"'  differences  of  the 
original  series. 

The  theorem  is  a  direct  interpretation  of  equation  (G3).  For,  if 
■^o""^"*  4)'"^"' ,  •  ■  •  .  are  all  small  in  comparison  with  z/i">,  then  the  ap- 
proxmiate  value  of     8,/"'     is     J/i"//o'"'. 

Corollary.  —  If  the  n"'  differences  of  the  given  series  are  con- 
stant, then  the  n*^  differences  of  the  new  series  are  also  constant,  and 
equal  to   m"    times  the  original    «-'*  differences. 

For,  if  J<"  is  constant,  j<"+i>,  z/<"+2>,  ....  are  all  zero,  and 
hence  (63)  gives,  rigorously, 


gC) 


/«».//'"' 


21.     To    illustrate   the   foregoing   results,   we    take    the    following 
table  of  cubes: 


T 

F(T)^T^ 

J' 

J" 

J'" 

100 
103 
106 
109 
112 
115 

1000000 
1092727 
1191016 
1295029 
1404928 
1620875 

+  92727 

98289 

104013 

109899 

+  115947 

+  5562 
5724 
5886 

+  6048 

+  162 

162 

+  162 

Here   the    interval     w  =  3.     If  we   take     m  :==  ^,     the   interval  is 
reduced  to  1,  and  hence  the  new  table  is  as  follows: 


T 

J3 

8' 

8" 

8'" 

100 
101 
102 
103 
104 
105 

1000000 
1030301 
1061208 
1092727 
1124864 
1157625 

+  30301 
30907 
31519 
32137 

+  32761 

+006 
612 
618 

+  624 

O  CD  CD 
+         + 

We   now   test  the  first   three  of  the   equations  (Gi) ;    substituting 


THE    THEORY   AND   PRACTICE    OF    INTERPOLATION.  35 

in  tlic  latter  m=:},  and  observing  that  the  differences  beyond  J'" 
vanish,  we  find 

S  /    I  /I  I X  /f  "  +     r,    ,1  III  S:  II    I  //  "  2      /  '"  Si  III    \    /I  '"  /R-'^i 

From  the  first  of"  the  above  tables,  Ave  take 

//;  =    +92727  z//'   =    +5502  JJ"  =    +162 

Whence,  from  (65),  we  derive 

8^'  =   30909-618  +  10   =   30301  8,/'  =  618  -12   =  606  8^"  =  6 

which  agree  exactly  with  the  values  found  in  the  second  table  above. 
It  will  be  observed  that  So'  and  S(,"  come  within  s'o  l^ail  of  equaling 
^JJ  and  '//„",  respectively;  while  8/"  =  ^\A,"',  exactly.  These  i-ela- 
tions  are  in  accord  with  Theorem  YII. 

22.  To  Exjiress  the  Differences  of  F{T)  in  Terms  of  the  (riven 
Functions  only.  —  Let  the  given  series  be  Fq,  F^,  F.^,  F^,  .  .  .  .  ;  then 
the  first  differences  are  i^i — F^,  F^ — F^,  F^ — F.^,  .  .  .  . ;  the  second 
difi'erences,  F.,  —  'IF,^F,,  F^  —  2F^^F,,....;  the  third  difter- 
ences,  F^  —  'dF,^'dF,—F,,F,  —  'iF.,^dF^—F,,....;  and  so  on. 
The    coefficients    evidently   follow   the    binomial    law.     Thus    we    have 

generally 

(66) 

Ji"'   =   F„-nF„_,+  '.^^^^F„_,-    .    .  +  (_iy„C,.i^„_,  ±  .    .  +  (_l)"-i„i^,+(_l)'.i^„ 

in  which,  according  to  the  usual  notation,  we  put  „CV  for  the  co- 
efficient of  x''  in  the  expansion  of  (l-\-x)'\ 

To  prove  (66),  let  us  assume  it  true  for  the  index  n;  then  the 
expression  for  the  ?i*  difference  immediately  folloMnng  /Ji;"  (i.e.,  j[">) 
will  be  obtained  by  increasing  the  subscripts  of  F„,  i^„_,,  ....  in 
(66)  by  unity.     We  therefore  have 

A"'  =  ^„+,-«^„+'-^^^^^„-i-  ■  .  +  (-iy+\,c,.+i^;-,-±  •  .  +  {~iyF,      (67) 


Subtracting  (66)   from   (67),  we  find 

+  {-ir'(„c,.^,+,A)K-r±  ■  .  .  +  (-i)"(«+i) /;+(-!)"+'/'„ 


jr"  =  A"  -  A"  =  ^,.+1  -  (« + 1)  ^,.  +  ^^^^  ^,,-1 


36 


THE    TIIEOEY    AND    PRACTICE    OF    INTERPOLATION, 


But,  as  proved  in  AlgL'l)ra,  wo  have 


r.C'r+l  +  ,flr     —      „+lC',._^i 


and  hence  the  preceding  equation  becomes 

(68) 
^r"  =  ^,,+1  -  ("  +  l)^,.+  ^^^^,.-i-   •    •  +  (-l)^+'„+iC',.+,i^„..,.±  .    .  +  (-!)■■+■  2^„ 

It  follows  from  ((38)  that  if  the  law  expressed  in  (GG)  holds  for 
n,  it  also  holds  for  n-\-l.  But  we  have  seen  above  that  the  expres- 
sion is  true  for  n  ^  1,  2  and  3.  Hence  it  is  true  for  n  r=  4,  and 
so  on  indefinitely;  the  equation  (6G)  is  therefore  true  for  all  positive 
integral  values  of  n. 

23.  To  Express  Any  Function  of  a  Given  Series  in  Terms  of  Some 
Particular  Function  (Fq),  and  of  the  Differences  (f/o,  })„,  c^,  .  .  .  .) 
which  Follow  that  Function. — As  before,  let  F^,  F^,  Fn,  F-j,  .  .  .  . 
denote  the  given  series,  the  differences  being  taken  as  in  the  schedule 
below : 


F{T) 

J' 

J" 

J'" 

Jiv 

Jv 

J  VI 

K 

«0 

a, 

«3 
"4 

h.. 

'■l 

'o 

/o 

K 

K 

'■;! 

(L 

1 

^'l 

K  1 

K  " 

F 

«„-I 

i'   , 

^„+> 

ff„ 

Let  it  be  required  to  express  F^  in  terms  of     F^,  a^,  \,  Co,  d^, 
From  the  nature  of  the  differences,  we  have 


^1  = 

F.  = 

F.  = 


K  +  % 

J'\  +  a, 

F.,  +  a„ 


(F„  +  2a^  +  h^)  +  (a,  +  2b,  +  c^)    =   F,  +  3a,  +  3lr  +  r^ 


THE    THEORY    AXD    I'KAOTICE   OF   IXTEKPOT.ATION.  37 

and  so  on.     The  coefficients  a<^aiii  follow  the  binomial  law,  which  sug- 
gests for  the  form  of  tlie  general  term  — 

n(n-1)  ,        n{n-l)(u-2) 
K  =   /•;  +  ««„  +  -^^i„+^ ^ >  r^+    ....  (69) 

'Vo  })rove  (()9)  by  induction,  we  assume  that  it  is  true  for  the 
index  ii.     Moreover,  we  evidently  have 

We  may  now  find  o„  in  terms  o^f  a^,  h^,,  c^,  (Iq,  .  .  .  .  from 
(G9),  —  since  the  relation  is  here  the  same  as  the  relation  of  F,^  to 
i^oj  «oj  K,  Co,  •  •  •  •  ;     thus  we  obtain 

,        u(n  —  l) 
a„   =   a^+7ih^+ c„+    .... 

Adding  this  value  of  a„  to  that  of  i^„  given  by   (69),  we  find* 

Thus,  having  assumed  the  relation  (09)  to  be  true  for  the  index 
71,  we  find  by  (70)  that  it  is  also  true  when  n-\-l  is  written  for  n; 
but  we  have  shown  directly  that  (69)  holds  for  n  ==  1,  2  and  3. 
The  formula  (69)  is  therefore  true  for  all  positive  integral  values  of  n. 


*We  here  omit  the  proof  for  the  general  term,  since  the  process  is  the  same  as  in  §22. 

\ 


38 


THE  THEORY  AND  PRACTICE  OF  IKTBRPOLATION. 


EXAMPLES. 

1.  Tabulate  the  five-place  logarithms  of  25,  30,  35,  ....  65,  70, 
and  take  the  differences  to  the  fifth  order  inclusive.  Retain  a  copy 
of  the  table  for  further  use. 

2.  Tabulate  F{T)  =  log  cosT,  to  five  decimals,  for  T=  50°, 
53°,  56°,  ....  74°,  77°;  difference  to  the  fifth  order,  as  in  Example  1. 
Retain  a  copy  of  tlie  table. 

3.  Verify  the  accuracy  of  both  the  functions  and  their  ditferences 
in  Examples  1  and  2,  by  noting  the  degree  of  regularity  in  l\  accord- 
ing to  the  method  of  §8. 

4.  Also,  rigorously  check  the  differencing  in  the  above"  examples, 
by  taking  the  algebraic  siim  of  each  separate  order,  as  explained 
in  §3. 

5.  Add  the  two  series  of  functions  tabulated  in  Examples  1  and 
2  ;  difference  the  new  series  as  before,  and  see  that  the  resulting 
values  of  .r  are  the  sums  of  the  fifth  diff"erence8  of  the  other  series, 
according  to  Theorem  IV. 

6.  Correct  the  errors  in  the  following  tables  by  the  method  of 
differences: 


{a) 


(V) 


(S) 


T 

F(T)  =  \, 

0.21 

4.7G2 

.23 

4.348 

.25 

4.000 

.27 

3.704 

.29 

3.465 

.31 

3.226 

.33 

3.030 

.35 

2.857 

.37 

2.703 

0.39 

2.564 

Appa.  Alt. 

Mean 

of  Star 

Refraction 

o 

10 

5  19.2 

12 

4  27.5 

14 

3  49.5 

16 

3  18.4 

18 

2  57.5 

20 

2  38.8 

22 

2  23.3 

24 

2  10.2 

26 

1  58.9 

Latitude 

Reduction 

o 

1        ir 

0 

0     0,00 

2 

0  4S.(»2 

4 

1  35.80 

6 

2  23.12 

8 

3     9.75 

10 

3  55.11 

12 

4  40.05 

14 

5  23.28 

16 

6     4.95 

18 

6  44.86 

THE  THEOKY  AND  PRACTICE  OF  INTERPOLATION. 


39 


(cZ) 


(«) 


(/) 


T 

F(T)=.  Tsinr 

0.48 

0.7125 

.50 

.7173 

.52 

.7226 

.54 

.7273 

.56 

.7349 

.58 

.7419 

.60 

.7494 

.62 

.7568 

.64 

.7660 

.66 

.7751 

.68 

.7847 

.70 

.7947 

0.72 

0.8052 

Date 

Log.  Dist.  of 

1898 

Mars  from  Earth 

Sept.  17 

0.139162 

21 

.130819 

25 

.122145 

29 

.113130 

Oct.   3 

.103759 

7 

.094015 

11 

.083857 

15 

.073360 

19 

.062478 

23 

.051135 

27 

.039438 

31 

.027351 

Nov.  4 

0.014875 

Date 

Lunar  Dist.  of 

1898 

Jupiter 

Dec.  1.0 

105  5  59 

1.5 

99  18  28 

2.0 

93  31  31 

2.5 

87  44  46 

3.0 

81  57  48 

3.5 

76  10  17 

4.0 

70  21  14 

4.5 

64  30  37 

5.0 

68  39  44 

5.5 

52  42  5 

6.0 

46  43  12 

6.5 

40  40  43 

7.0 

34  34  29 

7.  Tabulate  the  following  rational  integral  functions  for  the  as- 
signed values  of  the  argument.  Before  taking  the  differences,  state 
at  which  order  the  latter  become  constant,  and  compute  the  constant 
value  in  each  case,  by  Theorem  V.  Then  take  the  differences,  and 
see  that  the  results  agree  with  the  comjjuted  values. 

(a)  F{T)  E   T''-  50 T'  +  100 T-. 

(Tabulate  for     T  =    -8,   -6,   -4,   -2,     0,   +2,    +4,   +6,   +8.) 

(b)  F(T)  E  ST''  -72' -400.     (T  =   8.0,  8.3,  8.6,  .    .    .    .9.8.) 
(e)     F{T)  E  0.16r<-0.3r-'.  (T  =  2,  3,  4,  5,  6,  7,  8.) 

8.  By  means  of  the  first  of  equations  (1),  compute  the  value 
of  //'  which  immediately  follows  log  cos  56°  in  the  table  of  Example  2. 
The  value  of  w  (=3°)  must  be  expressed  in  circular  measure.  Com- 
pai"e  the  computed  with  the  tabular  value. 

9.  Tabulate  F{T)  =  log  T,  to  five  places  of  decimals,  for 
T  =  .30,  40,  50,  60,  70;  denote  this  table  by  B,  and  that  of  Example  1 
by  A.  A  and  B  then  diifer  only  in  w,  the  interval  having  now  been 
doubled.  Then,  in  the  second  of  the  equations  (64),  put  in  =  2,  and 
substitute  from  A  the  values  of  JJ',  ,]^"',  zl^'\  and  j/,  which  correspond 
to  T  ^  40.  Whence,  compute  the  value  of  8/  corresponding  to 
T  ^  40  in  B,     and  compare  computed  with  actual  value. 

10.  In  Example  1,  compute  the  quantities  J„'''  and  i^5(=log50), 
by  (66)  and  (69)  respectively;  compare  the  results  with  the  values 
found  in  the  tabic. 


CITAPTER  II. 

OF     INTEKPOLATION. 

24.  Statement  of  the  Problem.  —  Given  a  series  of  munerioal  values 
of  a  function,  for  equidistant  values  of  the  argannent,  it  is  required  to 
find  the  value  of  the  function  for  any  intermediate  value  of  the  argu- 
ment, independently  of  the  analytical  foi'ni  of  the  function,  Avhich  may 
or  may  not  be  given. 

Interjiolation  is  the  process  or  method  by  which  the  rc(|uired 
values  are  found. 

Without  certain  resti'ictions  or  assumptions  as  to  the  character  of 
the  function  and  the  interval  of  its  tabulation,  the  problem  of  inter- 
polation is  an  indeterminate  one.  Thus  it  is  evident,  a  'priori,  that 
from  a  series  of  temperatures  recorded  for  every  noon  at  a  given 
station,  it  would  be  impossible  to  obtain  by  interpolation  the  tempera- 
ture at  8.00  P.M.,  for  a  given  ^?ij.  If,  per  contra,  the  thermometric 
readings  were  recorded  for  7.00,  7.10,  7.20,  7.30,  ....  p.m.,  it  is  highly 
probable  that  the  temperature  at  7.14  p.m.  could  be  intei'polated  with 
accuracy. 

The  Nautical  Almanac  gives  the  heliocentrjc  longitude  of  Jupiter 
for  eveiy  4th  day;  but,  because  of  the  slow,  continuous,  and  syste- 
matic character  of  Juinter^s  orbital  motion,  it  is  found  sufficient  to 
compute  the  longitudes  from  the  tables  direct  for  every  40th  day  only. 
The  intermediate  places  are  then  readily  interpolated  with  an  accuracy 
which  equals,  if  indeed  it  does  not  exceed,  that  of  direct  computa- 
tion. 

The  moon's  longitude  is  given  in  the  Nautical  Aluuoiac  for  every 
twelve  houi-s;  for  the  moon's  orbital  motion  is  so  rapid  and  compli- 
cated that  it  would  prove  inexpedient  to  attempt  the  interpolation  of 
accurate  values  of  the  longitude  from  an  ephemeris  given  for  whole 
day  intervals. 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


41 


It  therefore  appear.s  that,  to  render  the  problem  of  interpolation 
determinate,  the  tabnlar  interval  (w)  nms^t  be  ssnthciently  small  that 
the  nature  or  law  of  the  funetion  will  be  definitively  shown  by  the 
tabnlar  values  in  question.  The  eondition  thus  imposed  will  be  satisfied 
when,  in  a  given  table,  the  differences  become  either  rigoroui^} ij  w 
sensUAij  constant  at  some  paiticular  order.*  This  follows  from  the 
fact,  soon  to  be  proved,  that  for  all  such  cases  a  formula  of  interpo- 
lation can  be  established,  either  rigorously  or  sensihly  true,  according 
to  the  foregoing  distinction. 

25.  Extension  of  Formula  (69)  to  Fractional  and  Negative  Values 
of  a,  Provided  the  Differences  of  Some  Particular  Order  are  Constant. — 
We  have  shown  (Theorem  V)  that  the  differences  of  a  rational  inte- 
gral function  vanish  beyond  a  certain  order.  We  j^roceed  to  prove 
that,  for  any  such  function,  the  formula  (69)  is  rigorously  ti-ue  for  all 
values  of  n. 

Let  F(T)  denote  any  function  whose  differences  become  con- 
stant at  the  order  i,  and  let  j">  =  Ig-,  F(T)  and  its  differences  are 
then  shown  in  the  schedule  on  following  page. 


*  Excepting,  of  course,  any  periodic  function  whose  tabular   interval  (m)  differs   but   little  from 
some  multiple  of  its  period,  P.     An  example  of  such  a  series  is  the  following  : 


Date,  1S98 

Day  of  the 
Year 

Heliocentric 
Longitude 
of  Mercury 

J' 

J" 

Jill 

Jan.     4 
Apr.     4 
July     3 
Oct.     1 
Dec.  30 

4 

94 

184 

274 

364 

o         / 

93      0 
105    33 
117     40 
129     14 
140     10 

+12  33 
12  7 
11     34 

+10    56 

—26 
33 

-38 

1 

—7 
—5 

where  P  (the  time  of  one  revolution  of  Mercury)  =  87.97  days;  and  hence  M  =  90''  =  P  +  2''.03. 
The  differences  J'  therefore  correspond  to  a  tabular  interval  of  2.03  days,  and  not  to  the  interval 
90  days,  as  the  table  itself  would  indicate.  Now,  the  actual  value  of  Mercury's  longitude  for  Jan.  14 
is  found  from  the  Nautical  Almanac  to  be  I  =  149°  40';  if,  however,  we  fail  to  account  for  the 
periodic  character  of  this  function,  and  argue  solely  from  the  numerical  data  at  hand,  we  find  by  a 
rough  interpolation,  for  Jan.  14, 

;  =  93°.0  +  (U  X  12°.6)  =  94°.4 

which  bears  no  relation  to  the  truth.  The  pos.sibility  of  thus  committing  serious  error  through  fail- 
ing to  account  for  completed  periods  or  revolutions,  suggests  the  necessity  of  caution  in  this 
direction. 


42 


TIIE    THEORY    AND    PRACTICE    OF    INTERPOLATION. 


T 

F(T) 

J' 

J" 

j(i) 

t 

K 

t+u, 

F. 

% 

Ik 

t  +  2u, 

t  "T  OOJ 

«1 

a.. 

h 
b. 

'o 
'o 

h 

\ 
1 

'o 

i;+(i  +  2)<o 

i-V. 

/^..    !  •   •  • 

*+(;+3)u. 

J'%. 

''i+S 

From   (.'30)   wu  obtain,  in  succession, 


With  the  condition  assumed,  these  equations  give 

Hence,  in  tliis  case,  the  expansions  (0)  end  at  the  (i-|-l)th 
term.  It  follows  that,  under  the  present  assumption,  the  expansions 
(0)  are  valid;  in  other  words,  F(t-\-iioj)  is  capable  of  expansion  by 
Taylor's  Theorem  for  all  values  of  7t  within  the  limits  of  the  given 
table.     Hence,  for  all  such  values,  we  have 

F.  =  F(t  +  n»)  =  F{t)  +  n,.F'{t)  +  "^  F"{f)  +  .    .    .    .  +  ^^  i^'" (0  (71) 

E  LI 


Let  us  now  consider  the  expression 


n(n  —  V)  , 


•  + 


i(m-1) 


(w-i+1) 


(72) 


11      "  ■  li 

Substituting,  successively,       n  =^  0,  1,  2,  3,  ...  .  i  +  3,      in  (72),  we 
•get,  according  to  (69), 

Q  =   F^,  F,,  F,    F„  ,  .  .  .  2^,+3,     respectively. 

Substituting    these    same  values  of  n  in   (71),  we    evidently  obtain  the 
same  results,  namely  — 


F„  =   Fg,  i''i,  F^,  F^,  .    .    .    .  /''j+3,     in  succession. 


THE    THEORY    AND    PRACTICE    OF    INTERPOIiATION.  43 

Hence,  i^„  and  (^  jirc  eqnal  to  each  other  for  more  than  i  valnes  of 
n.  But  Fn  and  Q  are  l)oth  expressions  of  the  degree  i  in  n.  Now, 
when  two  expressions  of  the  degree  i  in  n  ai"e  equal  to  each  other  for 
more  than  /  vahies  of  w,  they  are  equal  foi-  all  values  of  n.  There- 
fore, for  all  values  of  n,  fractional  and  negative,  we  have 

F„  =  Fit+n.)    =   i<;+  ua,+  '^  ^+    .    .    .    .  +  '^''''^  '    '    '  (-'+^^  ,„       (73) 

11  LI. 

pi-ovided  that     J  "  ^  /„  ^     constant.     This  is  the  fundamental  formula 

of  interpolation,  and  is  known  as  Newton's  Formidu. 

26.  Second  Proof  of  Newton's  Formida,  for  Constant  Values  of 
^..1.  —  Formula  (73)  is  readil}'  jjroved  by  means  of  equation  (59),  in 
which  ni  may  have  any  value.  The  only  condition  necessary  for  the 
validity  of  (59)  is  that  the  expansions  (0)  are  themselves  valid.  But 
since  we  assume  that  the  differences  beyond  z/<''  vanish,  it  follows  (as 
proved  in  the  last  section)  that  the  expansions  (0)  are  valid.  Hence 
(59)  gives,  rigorously, 

.,  .,      mhn—\)     .  ,,  m,(m  —  \)  .    .    .  (m—i  +  \)    ^,., 

V    =    mJ<+      \        ^JJ'+    .    .     .    .+-^ 1 -^ ^J„<" 

II  LI 

From  the  definition  of  8o    (see  schedule,  p.  31),  we  have 

S;  =   F(t  +  mo>)-  F(t)   =   F„,-F^ 

.:      F„  E   F^t  +  mw)    =    -f'o  +  So' 

^  ,,      m(»i—l)    ^  ,,  m(m~l)    .    .    (vi—i+1)    ,   ., 

\L  Li 

which  is  the  same  as  formula   (73),  except  that    m    is  written  for  n. 

27.  To  Find  n,  the  Interval  of  Interjjolation.  —  The  binomial  co- 
efficients of  Newton's  Foi-mula  are  given  in  Table  I,  for  every  hun- 
dredth part  of  a  unit  in  the  argument  n.  The  quantity  ?t  is  called 
the  interval  of  Interjiolation,  and  in  practice  is  always  less  than  unity. 
To  obtain  an  expression  for  n,  suppose  that  we  are  to  interpolate  the 
value  of  the  fi;nction  corresponding  to  the  ai'gument  T,  whose  value 
lies  between    t    and    t-\-w;    then  we  shall  have 

F„  E  F{t  +  ni^)    =   F{T)     ,     or     t  +  no,   =    T 

and  therefore 

T  —  f 
n    =    "—^  (74) 

(0 

which  determines  the  interval    n. 


u 


THE  THEORY  AXD  PKACTICE  OF  INTERPOLATION. 


28.     Example.  —  From  the  following  tal)le  of    2'\   find  tlic  \i\\ul- 
of  {2.8y  by  Newton's  Formula: 


T 

F(T)  =  T* 

J' 

J" 

J'" 

Jiv 

Jv 

2 
4 
6 

8 
10 
12 
14 

16 

256 

1296 

4096 

10000 

20736 

38416 

+  240 

1040 

2800 

5904 

10736 

+  17680 

+  800  1  ,  „..., 
._„„  1  +  960 
1'60    .„,, 

3104 

4832  ,  .0119 
+  6944   ^--^J- 

+  384 

384 

+  384 

0 
0 

Here  avc  have 


T   =  2.8 

r,„  =  +240 

f   =  2 

/,^,  =  +800 

<„  =  2 

r^    =  +960 

.„   ^  2.8--J  _ 

0.4 

r/„  =  +384 

/':  =  16 

r      =      0 

It  will  be  convenient  to  denote  the  coefHeients  of  a„,  b(,,  c^,,  .  .  .  . 
in  (73)  by  A,  B,  C,  .  .  .  .  ,  respectively.  Then,  fi-om  Table  I  (with 
argmnent    n  =z  0.40),    or  by  direct  computation,  we  find 


We  therefore  obtain 


A  = 

+  0.40 

C 

=  +0.0G40 

B   = 

-0.12 

D 

=  -0.0416 

^0 

^ 

+  16.00 

Aa^ 

= 

+  96.00 

BK 

= 

-96.00 

Co, 

= 

+  61.44 

Dd, 

= 

-15.9744 

.-.   (2.8)^  =   7*;.4    =    +61.4656 

This  resnlt  is  easily  verified,  and  found  exact  to  the  last  figure. 
However,  since  Table  I  does  not  in  general  give  the  exact  mathe- 
matical values  of  the  interpolating  coefficients,  it  follows  that  functions 
interpolated  in  this  manner  cannot  ahvays  be  ahsolutehj  correct.  The 
results  may  be,  as  in  logarithmic  computation,  but  close  approximations 
to  the  truth. 

29.  Backward  Interj)olatio7i: — When  the  interval  of  interpolation 
approaches  unity,  it  is  usually  more  convenient  to  jjroceed  backwards 
from    the    function    which  follows    the    value    sought.       The    problem. 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


therefore,  is  to  find  /^_„;  for  this  purpose,  let  F (T)  he  differenced  :is 
in  tlie  schedule  below  —  the  values  of  //<"  being  supposed  constant  as 
before ; 


T 

F{T) 

J' 

J" 

J"' 

Jtv 

■  ■   ■ 

z](0 

t   —   3<D 

t  -2w 

t  —  w 

t 

t   +    0) 

t  +  2^ 

F-, 

F, 

F, 

a_„ 
(l-i 

Cto 
"i 

bu 

C-3 

c_„ 

d      r 

d  „ 

•     •     • 

0 
0 
0 
0 

C  +  Sw 

F, 

h„ 

"i 

d. 

«     •     • 

to 

AVe  might  substitute  — n  for  «  in   (73),  and  find  directl}', 

X,       .       s  (-n)(-n-l),        (-n)(-n-l)(-n-2) 


li 


11 


But  this  formula,  while  true,  is  inconvenient  from  the  fact  that  its 
coefficients  neither  converge  as  rapidly  as  the  binomial  coefficients  for 
-\-n,  nor  can  their  numerical  values  be  taken  from  Table  I.  To  avoid 
the  negative  interval,  we  have  only  to  suppose  the  series  inverted, 
thus  making  F-^  the  first,  and  F_^  the  last  of  the  tabulai-  functions. 
Then,  by  Theorem  III,  the   signs   of     //',  J'",  d",  .   .   .   .      are   changed, 


while  the  signs  of     J",  J",  . 


are  imaltered.     Now  the  value  of 


F_n  is  obtained  by  interpolating  forioard  with  the  interval  -\-n  in  the 
inverted  series;  hence  the  dift"erences  to  be  used  in  iS^ewton's  Formula 
are  — 

—  rt_, ,    +b_„,   — c.s,    +d^, 

We  therefore  have,  by  (73), 


F_,S  F(t-7i<o)^F^-na_,+ 


(75) 
7i(n-l)  J         w(w-l)(w-2)  7i(n-l)(7i-2)(n-3) 

HE  LL 

the  differences  being  taken  as  in  the  above  schedule.     The  coefficients, 
as  before,  are  taken  from  Table  I  with  the  argument  n. 

An  immediate  and  important  application  of  (75)  is  in  finding  the 
value  of  a  functtion  near  the  end  of  a  given  series.  Thus,  in  the  pre- 
ceding schedule,  suppose  the  series  ended  with  F^^,  and  it  were  required 
to  interpolate  a  value  of  J^  between  F_i  and  i^o-  since  the  differences 
&_i,  c_i,  f/_, ,  ....     (required  in  interpolating  forward  from  F^-i)  are  not 


46  THE    THEORY   AND   PRACTICE    OF   INTERPOLATION. 

given  ill  this  case,  the  formiUa  (75)  nnist  be  used;  n  being  the  inter- 
val of  the  required  function  from  i^,  toward  F_, . 

ExAiviPLE. —  From  the  table  of  T'^  given  on  page  44,  find  the  value 
of  (13.2(i)^ 

Taking     f  =  14,     we  find 

l-t-13.2C 
>,   =  2 =  0.3* 

which  is  the  interval  counted  hachwards  from  F  =  38416.  Hence, 
from  Table  I,  we  obtain 

A  =    +0.37  C  =    +0.06333 

B  =    -0.11655  D  =    -0.04164 

And  for  the  differences  required  by   (75),  we  have 

a_i   =    +17680  c_3  =    +2112 

b_„   =    +   6944  d_,  =    +   384 

Therefore,  by  (75),  we  derive 

F^  =  +38416.00 

-Aa_^  =  -   6541.60 

+  Bh_^  =  -     809.32 

-Cc_^  =  -     133.75 

+  Dd_^  =  -       15.99 

.-.  F„  =   (13.26)'  =  +30915.34 

By  direct  calculation,  we  find 

(13.26)'  =  30915.34492  + 

30.  Application  of  Newton's  Formula,  wlien  the  Differences  Be- 
come only  Approximately  Constant.  —  We  have  proved  (§§25  and  26) 
that  (73)  is  true  for  all  values  of  n,  provided  the  differences  of  some 
particular  order  are  rigorously  constant.  We  now  propose  to  show 
that,  if  the  value  of  n  lies  between  0  and  -|-1,  the  formula  is  very 
approximately  true  for  the  more  frequent  case  in  which  the  differences 
of  some  oi'der  become  approximately,  but  not  absolutely  constant.  The 
example  given  on  page  8  is  typical  of  this  case;  the  numbers  involved 
are  not  the  true  mathematical  values  of  the  quantities  represented,  and 
hence  the  irregularities,  as  already  explained. 

Let  Fo,  F^,  F^,  Fi,  ....  Fr ,  ...  .  denote  a  series  of  approxi- 
mate  tabular   values   of  any   fimction     F(T),     given   for   equidistant 


THE    THEORY    AND   PEACTICE    OF   INTERPOLATION. 


47 


values  of  T,  and  true  to  the  nearest  unit  of  their  last  figure;  let 
F(^,  Fi,  F^i  F3,  ....  Fr,  .  .  .  .  denote  the  corresponding  true  mathe- 
matical values  of  the  series,  which  we  shall  designate  generally  as  F; 
also,  let  Fr  ^  F,.-\-f^;  /".  being  the  difference  between  the  true  and 
approximate  values,  due  to  the  omission  of  decimals  in  the  tabular 
quantities. 

The  differences  of  F,  and  those  of  the  series    fo,  f\ ,  f-i,  fa,  •  •  •  •  ■, 
are  now  defined  by  the  two  schedules  l)elow: 


T 

F(T) 

J' 

J' 

J'" 

■   ■ 

JiO 

Jc+ii 

•       ■ 

t 

«+    (0 

t  +  2u> 
f +  3(0 
t  +  4(0 
t  +  5(0 

Fi 
F, 
Fs 
F, 
F, 

"2 
a. 

''0 
h 

^0 

c„ 

"3 

"'0 

"'-2 

. 

(A) 


T 

/ 

J' 

J" 

J'" 

•  • 

j(i) 

JC+l) 

•   • 

t 

/o 

t    +    (0 

/, 

«0 

A, 

t  +  2<o 

f. 

«1 

A, 

Vo 

K 

t  +  3(0 
t  +  4(0 

«3 

ft 
ft 

yi 

Mo 

Ml 

t  +  5(0 

A 

«4 

ft 

ys 

/x.. 

(B) 


Then,  since      F  =z  F-{-f,      it  follows  from   Theorem  IV  that  the 
differences  of  F  are  as  given  in  the  appended  table  : 


T 

^(T) 

J' 

J' 

J'" 

.     . 

J^<> 

JC+l) 

•   • 

t 

t   +    (0 

it +  2(0 
t  +  3(0 
i  +  4(o 
«  +  5(0 

^0  =  ^o+/o 

^^2    =    -F'2+/2 

ao  +  «o 
«i  +  «i 

«o  +    (C, 

as  +  «3 
a,  +  «, 

^'„  +  ft 
/',  +  ft 
L  +  /3, 

i3  +  ft 

/'.  +  ft 

'•u  +  yo 
''1  +  yi 

C„  +  y„ 

<-s  +  ys 

"'0  +  Mo 

W,  +  fXi 

W.,  +  /i„ 

' 

Let   us   now   suppose   that   the   differences  j<'+i'  in    Table  (C)  are 
either  alternately  -|-  and  — ,  or  that  -|-  and  —  signs  follow  each  other 


48  THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 

irregiilnrly.     ^Moreover,  tlu'  lorcgoing  definition  of  F  requires    that  the 

♦ 

terms  in  //<■+"  are  sufficiently  small  to  indicate  that  no  ei-rors  exceed- 
ing half  a  unit  in  the  last  place  exist  in  the  functions  F(^T).  The 
values  of  //''  are  then  approximately  constant,  and  thei-efore  Table  (C) 
represents  the  typical  case  in  practice.  We  pi-oceed  to  investigate  the 
accm'acy  of  Newton's  Foi-nuila  as  applied  in  this  case;  assuming  that 
a  is  always  taken  within  the  limits  0  and  -|-1,  and  that  terms  beyond 
J<"  are  neglected. 

Applying  (7)})  to  find  F,^  from  Table  (C),  and  omitting  the  terms 
beyond  //<•>,  we  have 

F„   =    (^,+/o)  +^K  +  «„)  +  i?(/>o  +  /?„)  +  (7(o,+y„)+    .    .    .    .+i(/„  +  \,)  (76) 

in  which     A,  B,  C,  .  .  .  .  L     denote  the    binomial   coefficients   of   the 
//th  order.     Let  us  now  examine  the  approximate  formula  (76),  to  dis- 
cover its  maximum  ei-ror  when  all  conditions  conspire  to  that  end. 
The  formula   (7(5)  may  be  written 

F.   =    {F,  +  Aa,+  Bh,+  ....  +LI„)  +  Q',+Aa,+Bp,+  ....  +  LK)       (77) 

For  brevity,  let  us  put 

Q  E  F^  +  Aa,-^Bh^+  ....  +LI, 

Ji  E  fo  +  Aa„  +  BI3^+  .    .    .    .  +7>A„        }■  (77«) 

.-.  F„  =  Q  +  li 

It  will  be  observed  that  Q  is  the  value  obtained  for  i^„  when 
(73)  is  applied  to  Table  (A),  terms  beyond  j"'  being  neglected.  We 
leave  the  discussion  of  Q  for  the  present,  to  consider  the  quantity  M, 
which  evidently  expresses  the  error  of  interi)olation  due  to  the  un- 
avoidal)le  eri'ors,  f,  contained  in  the  tabular  functions  F. 

Applying  tlio  fornnilac  of  §22  to  the  differences  of  Table  (B), 
we  have 

«o    =  /i  — /o 

A  =  A-2A+f„ 

y<>  =/3  — 3/3+  3/,  — /„  , 

80  =/,-4,/;+  Gf,-  iA+f„  I  ^'  > 

*o  =  /o  -  5/i  + 10/3  -  10/,  +  5/;  -  /„ 


THE    THEOEY   AND   PRACTICE    OP   INTERPOLATION.  49 

Hence,  from  (77a),  we  obtain 

B   =  /;  +  Aa^  +  Blio+  Cy„  +  m„  +  Ec„+   .    .    .    .  +L\„ 

=  /o  +  A  (/;-/;,)  +  i?(/,-2A+/„)  +  C(/,-3/;+3/,-/„) 
+  7;(/,-4/3+6/;-4/,+/„)+ .  .  .  . 

+f„(B-3C+6D-10E+  .    .    .    .)  +f^(C-AD  +  10E~  .    .    .    .)  \     (79) 

+f,(D-5E+  .    .    .    .  )  +ME-  ....)+....  J 

Now  the  binomial  coefficients     A,  B,  C,  .  .  .  .     are  connected   by 
the  following  relations: 


Hence,    since    we    have    assumed   that   n   lies    between    0    and    -(-1,   it 

follows   that      A,  B,  C,  .  .  .  .      are    alternately  positive   and   negative, 

thus: 

ABODE         .... 

+        -        +        -         +  .... 

We  therefore  draw  the  following  conclusions  respecting  (79) : 

The  coefficient  of    /;     is     +  ; 
«  (I  <</•<<     . 

"    A    "    + ; 
((         «        li    -f    li    . 


Now,  since  the  values  of  F  are  supposed  true  to  the  nearest  unit 
of  the  last  decimal  figure,  the  quantities  /'  may  have  any  value  between 
— 0.5  and  -\-0.5,  in  tei'ms  of  the  same  unit;  hence,  it  follows  from  the 
foregoing  conclusions  that  if  we  take 

/,   =    +0.5        /,  =    -0.5        ./;  =    +0.5        /,   =    -0.5  ....  (80) 

the  sum  of  all   the  terms   after  the  first  in  the   right-hand   member  of 
(79)  will  be  numerically  a  maximum,  with  the  -|-  sign. 

We  shall  now  show  that  the  coefficient  of  /J,  in  (79)  is  a  positive 
number.     For  this  pupose,  let  us  consider  the  identity 

(l-a;)-'(l-x)"  E   (l-x)"-> 

which,  for  all  values  of  x  numerically  less  than  tmity,  may  be  expanded 
into  the  form 

(l+a;+a-2+.r^+  .    .  +x<+  .    .  )(1 -Ax  +  Bx^- Cx^+  .    .  ±Lx':f  ■    ■)  =  (1 -«)""' 


50  THE   THEORY   AJSTD   PIU.CTICE    OF    INTERPOLATION. 

Upon    equating    the    coefficients    of   .«'   in    the    two    members    of  this 
identity,  we  find 


1-A+B- C+ 


±L  =  f-lV      {n-l)in-2){n-Z)  .    .    .  (n-{) 


=  (i-i)  1-lVi-'^ 


.     1 


^ 


Now,  the  first  member  of  this  equation  is  the  coefficient  of /o  iii 
(79) ;  and  since  the  final  member  contains  only  positive  factors,  it 
follows  that  the  coefficient  of  Jl  in  (79)  is  a  jjositi.ce  quantity.  Ac- 
cordingly, if  we   take    fo  =  -{-0.5,     in   conjunction  with  the  values  of 

f\i  fti  fii designated  in  (80),  the  value  of  R  given  by  (79) 

will  then  be  the  greatest  possible  under  the  assigned  conditions. 

We  now  append   a   table    of  the  quantities    fo,  f\,  fi,  fz-, 

as  above  determined,  with  their  differences  : 


T 

/ 

J' 

J" 

J'" 

Jiv 

Jv 

Jvl 

Jvil 

t 

t  +  w 
)!  +  2a. 
t  +  3a) 
!'  +  4«, 

<  +  5(u 

+0.5 

+  0.5 
-0.5 
+  0.5 
-0.5 

+  0.5 

0.0 

-1.0 
+  1.0 
-1.0 
+  1.0 

-1 

+  2 
—2 
+  2 
_2 

+  3 

-4 
+  4 
-4 

-7 
+  8 
-8 
48 

+  15 

-16 
+  16 

-31 

+  32 
-32 

+  63 

-64 

(B') 


The    special    values    which    must    be    assigned    to    the    quantities 
/o,  ao,  /8o,  7o,  •  •  •  •     of  Table  (B)   are,  therefore. 


/o  «o  ^0  7(1 

+0.5         0.0         -1  +3 


+  15 


in    units   of   the   last   place   of  the   tabular  quantities  F.     Substituting 
these  values  in  the  original  expression  for  li  given  in  (77a),  namely. 


we  obtain 


B   =  f,  +  Aa,  +  Z?/3„  +  (7y„  +  .    .    .    . 


B  =    +0.5-/i  +  3C-7Z>  +  15JS'-31i^+63G - 


(81) 


which  gives  the  maximum  value  possible  to  R  for  j^^J. 


THE    THEOKY   AND   PEACTICE    OP   INTERPOLATION. 


51 


To   evaluate  (81)  for  different  values   of   n   between   0    and   -f-l, 


we  make  use  of  the  following  abridged  table: 


n  =  A 

B 

c 

D 

E 

F 

G 

+ 

— 

+ 

— 

+ 

— 

+ 

O.OO 
O.IO 
0.20 
0.30 
0.40 
0.50 
0.60 
O.70 
0.80 
0.90 
l.OO 

.0000 
.0450 
.0800 
.1050 
.1200 
.1250 
.1200 
.1050 
.0800 
.0450 
.0000 

.0000 
.0285 
.0480 
.0595 
.0640 
.0625 
.0560 
.0455 
.0320 
.0165 
.0000 

.0000 
.0207 
.0336 
.0402 
.0416 
.0391 
.0336 
.0262 
.0176 
.0087 
.0000 

0000 
0161 
0255 
0297 
0300 
0273 
0228 
0173 
0113 
0054 
0000 

.0000 
.0132 
.0204 
.0233 
.0230 
.0205 
.0168 
.0124 
.0079 
.0037 
.0000 

.0000 
.0111 
.0169 
.0190 
.0184 
.0161 
.0129 
.0094 
.0059 
.0027 
.0000 

+ 

— 

+ 

— 

+ 

— 

+ 

(D) 


From  these  values  we  tabulate  as  follows  : 


n 

0.00 

0.10 

0.20 

0.30 

0.40 

0.60 

0.60 

0.70 

0.80 

0.90 

1.00 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

-     B 

.000 

.045 

.080 

.105 

.120 

.125 

.120 

.105 

.080 

.045 

.000 

+   3C 

.000 

.085 

.144 

.178 

.192 

.187 

.168 

.136 

.096 

.049 

.000 

-  ID 

.000 

.145 

.235 

.281 

.291 

.274 

.235 

.183 

.123 

.061 

.000 

+  15^ 

.000 

.241 

.382 

.445 

.450 

.409 

.342 

.259 

.169 

.081 

.000 

-31i^ 

.000 

.409 

.632 

.722 

.713 

.635 

.521 

.384 

.245 

.115 

.000 

+  63  (? 

.000 

.699 

1.065 

1.197 

1.159 

1.014 

.813 

..592 

.372 

.170 

.000 

If,  now,  we  let  R^i  Rzi  -^4?  • 
differences  beyond  the  2d,  3d,  4th, 
lected,  then,  from  (81),  we  find 

R,  =  0.5  -  i? 


denote  the  values  of  B  when 
.    order  respectively  are  neg- 


i?3  =  0.5-5  +  ZC 

B,  =  0.5-B  +  3C-7D 


(82) 


From  the  last  table  we  obtain,  by  successive  additions,  the  values 
of  R.2,  R3,  Ri,  .  .  .  .  as  defined  by  (82);  these  values  are  tabulated 
below  : 


n 

0.00 

0.10 

0.20 

0.30 

0.40 

0.50 

0.60 

0.70 

0.80 

0.90 

1.00 

R-i 

0.50 

0.55 

0.58 

0.60 

0.62 

0.63 

0.62 

0.60 

0.58 

0.55 

0.50 

B, 

0.50 

0.63 

0.72 

0.78 

0.81 

0.81 

0.79 

0.74 

0.68 

0.59 

0.50 

B, 

0.50 

0.78 

0.96 

1.06 

1.10 

1.09 

1.02 

0.92 

0.80 

0.66 

0.50 

^5 

0.50 

1.02 

1.34 

1.51 

1.55 

1.50 

1.37 

1.18 

0.97 

0.74 

0.50 

•Be 

0.50 

1.42 

1.97 

2.23 

2.27 

2.13 

1.89 

1.57 

1.21 

0.85 

0.50 

B, 

0.50 

2.12 

3.04 

3.43 

3.43 

3.14 

2.70 

2.16 

1.59 

1.02 

0.50 

52  TIIE    THEORY   AIS^D   PRACTICE    OF   INTERPOLATION. 

Whence   it   is   seen   that  the   greatest  jwssible  vahies  of  B,  under 
the  assumed  conditions,  are  — 


'J 


Ii„ 

i?3 

R, 

^5 

^s 

i?, 

0.6 

0.8 

1.1 

l.G 

2.3 

3.4 

(83) 

While  it  is  obvious  that  the  combination  of  accidental  errors  /, 
shown  in  Table  (B'),  is  very  improbable,  yet  approximations  to  such 
combination  will  occur  occasionally  in  jiractice.  In  such  cases  the 
errors  (H)  in  functions  interpolated  by  Newton's  Formula  may  be  a 
considerable  part  of  the  values  given  by  (83).  These  values  show 
that  when  the  diflferences  beyond  J^  are  neglected,  the  error  i?  cannot 
be  greater  than  1.6,  in  units  of  the  last  place  in  J^.  In  all  probability 
this  error  will  not  exceed  one  unit ;  and  when  it  is  considered  that 
the  results  of  an  average  logarithmic  computation  are  uncertain  with- 
in this  amoimt,  we  are  justified  in  neglecting  the  error  I^,  provided 
that  fifth  differences  are  practically  constant. 

Beyond  ^5,  the  limiting  values  of  i?  increase  rapidl}-.  AVe  there- 
fore conclude  that,  aside  from  the  inconvenience  involved,  it  is  im- 
practicable to  interpolate  by  Newton's  Formula  when  the  differences 
beyond  J"  are  too  large  to  be  neglected.* 

We  now  consider  the  expression   Q  of  (77a),  that  is  — 

Q=  F,+  Aa,  +  Bh,+  .    .    .    .  +L1,  (84) 

Now,  because  the  differences  of  F  in  Table  (C)  become  approxi- 
mately constant  at  J'",  notwithstanding  the  irregularities  they  contain; 
so,  a  fortiori,  must  the  differences  of  F  in  Table  (A)  become  sensibly 
constant  at  J"',  the  quantities  of  this  table  being  mathematically  exact. 
Hence  the  differences  //<'+"  in  Table  (A),  namely, 


tn„ ,  m, ,  111, ,  in„ 


will  form  a  series  of  continuous^  but  very  small  terms,  whose  values 
are  nearly  equal  to  each  other.  Per  contra,  we  have  assumed  that  the 
differences 

♦Excepting  the  case  where  F{T)  is  a  rational   integral  function  of  T,  whose  tabular  values  are 
mathematically  exact. 


TIIE    THEORY   AND   PRACTICE    OF   INTERPOLATION.  53 

of  Table  (C)  either  are  alternately  -|-  and  — ,  or  that  -f-  and  —  terms 
succeed  each  other  irregularly.  It  follows  that  the  quantities  in  must 
be  numerically  less  than  the  maximum  value  of  /u,  in  the  series 

/^o>      /^i>      /^a)      /^a;        .... 

For,  otherwise,  if  the  quantities  m  exceeded  the  greatest  of  the 
quantities  fx,  the  former  would  mask  the  effect  of  the  latter  in  the  com- 
bined series  ?M-)-/i,;  hence  there  would  be  no  general  alternation  of 
signs  in  the  series 


+  /i„ ,     /// ,  +  jUi ,     //;„  +  ^2 


But  this  is  contrary  to  our  assumption  that  the  differencing  in 
Table  (C)  has  been  carried  to  an  order  z:/('+"  which  does  exhibit  a 
general  alternation  of  signs.  We  therefore  conclude  that  -m^  is  numeri- 
cally less  than  the  maximum  value  of  jx. 

I^ow,    from    Table  (B'),   we    observe    that    under    the    conditions 

assumed, 

The  maximum  value  of     a(=J')  is  1   =  (2)"; 

/8  (=-'")  "  2   =  (2y; 

y(=J"')  «  4   =  (2/; 


Hence,  itia  is  numerically  less  than  2'. 

We  have  observed  above  that,  as  a  consequence  of  the  conditions 
herein  assumed,  the  differences  of  F  in  Table  (A)  are  converging,  being 
practically  insensible  beyond  z/'";  hence  the  fundamental  expansions  (0), 
and  all  relations  deduced  from  these,  are  valid  in  this  case.  The  formula 
(59)  is  therefore  applicable  to  the  series  F'(T);  hence,  writing  u  for  m 
in  (59),  we  have 

8„'  =   Aa„ -\- Bl>o  +  Ccg  +  ....  +Ll^  +  31>ii„  +  Nn^+  .... 

in  which  as  many  terms  should  be  retained  as  accuracy  requires. 
But  we  also  have* 

and  therefore 

F„  =  F,  +  Aa„+Bb^+Cc^,+  ....  +Ll^  +  Mm^,+  Nn,+  .... 
*  See  §26,  where  the  same  relations  were  similarly  employed. 


54  THE   THEORY   AXD    PRACTICE    OF   INTERPOLATION, 

Now,  by  (84),  this  equation  may  be  written 

F„   =    Q+  Mm,  +  Nn,  +  .    .    .    . 

or 

F„  -  Q  =  Mm,  +  iV?io  +   .    .    .    .  (85) 

The  series  3//»|,  -|-  JVy^^  -]-....  therefore  expresses  the  diffei-- 
ence  between  the  true  mathematical  vahie  of  the  interpolated  function 
and  its  approximate  value  Q.  But  since,  as  above  observed,  the  differ- 
ences 7)1  are  nearly  constant,  it  follows  that  the  differences  n  are  small 
in  comparison.  Hence,  iV/?o  is  small  as  compared  Avith  3I))i,  ;  in 
brief,  ^Im^  represents,  very  nearly,  the  value  of  the  rapidly  converging 
series  3Imo  -\-  J\lio  -)-....  in  the  right-hand  member  of  (85) .  The 
latter  equation  may  therefore  be  wi'itten,  Avithout  sensible  error, 

F,,-Q   ^   Mm.,  (8(0 

From  (82)  we  dei'ive 

R^-R„  =  +  ZC  =  (2=-l)(+C) 
B^-R^  =  -  ID  =  (2^-1)  l-D) 
R,-R,^   +15E  =  (2^-1)  {  +  E)  \  (87) 


Ri+,-B,  =   {2'-l){-iyM 
From  the  last  of  these,  we  obtain 

±  2'M  =  R,+,  -  Ri  ±  M  (88) 

"We  have  shown  above    that   m^   is    numerically  less    than  2';    tliis 
condition  may  be  expressed  in  the  form 

m,  =   2'  sin  6 

where  6  may  have  aiuj  value   between    0    and  "lit.     From   this    relation 
we  obtain 

Mm.,  =   2'J/sme 

or,  by  (88), 

Mm.,  =   (7i',+i-i?,±Ji)  sine  (89) 

Substituting  this  value  of  Miuq  in   (86),  we  get 

F„- Q  ==  {R,^^-R,±M)  sine  (90) 


THE  THEORY  ANT)   PRACTICE  OF  INTERPOLATION.  55 

From  (77rt),  we  have* 

K~Q  =  i?.  (91) 

which,  subtracted  from  (90),  gives 

F,,  —  -f;  =   A'i+i  sin6'  -  (1+  siu  6)  A\±]H  siufii 

From  Table  (D)  above  we  see  that  beyond  J'"  the  coefficient  M 
cannot  exceed  O.O-i,  which  is  an  inappreciable  quantity  in  the  present 
discussion;    we  therefore  write  the  last  equation 

F„-F„  =  ^,+1  sin  e-  (1  +  sin  6)  E,  (92) 

The  quantity  i?i_^.i  is  numerically  greater  than  7?;,  and  both  are 
alike  in  sign;    this  condition  may  be  expressed  by  the  relation 

in  which  xp  has  a  definite  value  depending   upon  the  value  of  i.     Sub- 
stituting this  expression  for  i?j  in  (92),  the  latter  becomes 

K-Fn  =   i^i+i[sin6l- sinV(l  +  sine)] 

or 

K-Fn  =  ii',+i(sin0  cosV-sin'-.//)  (93) 

Since  cos^i/;  is  necessai'ily  positive,  and  — sin^i//  negative,  it  fol- 
lows that  the  coefficient  of  i?;+,  in  (93)  will  be  numerically  a  maximum 
when  sin  ^  attains  its  greatest  negative  value;  that  is,  when  d^'^ir. 
Taking    6  :=  %  tt    in   (93),  we  have 

F„  —  F^  =  ^;+i(-eos-f-sin-./^)   =    --K,+i  (94) 

which  is  the  maximum  numerical  value  possible  to    F^  —  F',i,    all  con- 
ditions favoring. 

If^j,  is  the  true  mathematical  value  of  the  required  function.  JT^  is 
the  approximate  value  of  this  quantity  which  is  obtained  by  applying 
Newton's  Formula  to  Table  (C),  neglecting  differences  beyond  ^''>: 
it  being  assumed,  (1)  that  the  given  functions  i^'o?  -^n  -^25  F'sj  •  •  •  • 
are  true  to  the   nearest  unit   of  their  last  digit;    (2)  that  n  is  positive 


*  The  quantity  R  defined  in  (77a)  is  not  distinguislied  by  a  subscript  in  tlie  earlier  part  of  this 
discussion.  Considered  as  a  particular  term  of  the  series  ^2,-^3,-^4,  •  •  •  ■  ,  however,  it  is  evi- 
dent that  R  should  be  designated  as  Ri . 


56  THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 

and  less  than  unity;  (3)  that  tlie  differences  //"•  are  approximately 
constant;  and  (4)  that  the  dift'erences  J"+'>  are  quite  small,  Avitli  -|- 
and  —  signs  following  irregularly.  Under  these  conditions,  it  follows 
from  (94)  that  the  computed  value  F^  t'l"  never  differ  from  the  true 
value  F„  by  more  than  the  quantity  /i*,.!., . 

One  point  further,  however,  must  be  considered.  In  computing 
Fn  by  (TG),  we  shoidd,  in  practice,  obtain  the  values  of  tlic  several 
terms  to  one  or  two  decimals  further  than  are  given  in  F,  to  avoid 
accumulation  of  errors  in  the  final  addition.  But  in  writing  the  sum, 
J^„,  the  extra  decimals  are  drojjped,  the  result  being  taken  to  the 
neared  tn/if,  as  in  F.  Thus  we  actually  use,  not  the  quantity  F„  ob- 
tained rigorously  by  (76),  but  a  close  approximation  to  that  value, 
which  Ave  may  denote  by  {F„).     Accordingly,  the  relation 

F„~  (i-;)  =   ±0.5 

expresses  the  maximum  discrepancy  between  F„  and  (F„).  Combining 
this  expression  with   (94),  we  finally  obtain 

F,_  _(/.;)   =    -i?,^,  ±0.5  (95) 

The  quantity  ^,^.1  +  0.5  therefore  represents  the  final  limit  of  error 
in  the  value  of  an  interpolated  function,  in  units  of  the  last  decimal 
of  F.  From  the  value  of  i?o  given  in  (88),  we  find  that  when 
j^-  is  nearly  constant,  the  limiting  error  is  +2.8  units.  Since  it  is 
highl}'  impi'obable  that  all  the  necessary  conditions  will  conspire  to 
produce  this  inaxhnvm  error,  we  may  add  that  when  the.  differences 
practically  terminate  at  the  fifth  order,  interpolated  functions  will 
occasionally  be  in  error  by  one  unit,  only  rarely  in  error  by  two  units, 
and  never  by  three. 

With  sixth,  seventh,  or  higher  differences  employed,  the  results 
become  subject  to  errors  which  in  most  cases  would  be  intolerable, 
and  which  would  probably  be  obviated  by  a  direct  calculation  of  the 
function. 

From  the  foregoing  investigation  it  therefore  appears  that,  for 
purposes  of  interpolation,  tabular  functions  should  always  be  given 
with  an    interval    sufficiently  small   that  differences   beyond    p  may  be 


THE    THEORY   AND   PRACTICE    OF    LNTERPOLATION. 


57 


neglected.  This  condition  is  generally  fulfilled  in  practice.  As  already 
stated  in  §24,  the  longitude  and  latitude  of  the  moon  are  given  in  the 
Nautical  Almanac  for  every  twelve  hours;  from  the  values  thus  given, 
intermediate  positions  can  always  be  safely  interpolated  by  using  diffei- 
ences  no  higher  than  the  fourth  or  fifth  order.  On  the  other  hand,  a 
table  of  the  moon's  longitude  for  every  24  hours  would  yield  diifer- 
ences  of  the  eighth  or  even  ninth  order;  the  use  of  which  in  Newton's 
Formula  might  produce  an  error  of  several  units  in  an  interpolated 
position. 

In  all  that  follows,  we  shall  assume  that  differences  beyond  the 
fifth  order  may  be  neglected.  This  assumption  made,  it  follows  from 
the  preceding  investigation  that  the  fundamental  formulae,  (73)  and 
(75),  may  be  applied  in  all  cases  without  sensible  error,  provided  that 
n  is  taken  less  than  unity. 

31.  We  shall  now  solve  an  example  which  illustrates  the  main 
points  of  the  foregoing  discussion.     If  we  tabulate  the  function 


F{T)    _    75555 


606607.920 


199841.772  T  +  50804.968  T'- 


+     5645.715  T^-      2169.395  T^  +      116.817  T^  +  1.507  T^ 


(96) 


for  T  =  0,  1,  2,  3,  ...  .  9,  we  find  that  the  true  mathematical 
values  terminate  in  the  fifth  decimal.  These  values  of  F {T)  are 
given  in  the  table  below,  with  their  differences: 


F{T) 


0 

8.42511 

1 

6.40508 

o 

5.89492 

3 

6.53508 

4 

7.66492 

5 

8.55508 

6 

8.65492 

7 

7.85503 

8 

6.76481 

9 

7.00512 

-2.02003 
-0.51016 
+  0.64016 
1.12984 
0.89016 
+  0.09984 
-0.79989 
-1.09022 
+  0.24031 


+  1.50987 

1.150.32 

+  0.48968 

-0.23968 

0.79032 

0.89973 

-0.29033 

+  1.33053 


J"' 


-0.35955 

0.66064 

0.72936 

0.55064 

-0.10941 

+  0.60940 

+  1.62086 


Jiv 


-0.30109 
-0.06872 
+  0.17872 
0.44123 
0.71881 
+  1.01146 


J^ 


+  .23237 
.24744 
.26251 

.27758 
+  .29265 


Jvi 


+  .01507 
.01507 
.01507 

+  .01507 


(A') 


This  table  corresponds  to  Table  (A)   of   the  last   section.     It  will 
be  observed  that  the  values  of  F  are  jjeculiar  fi-om  the  fact   that   the 


58 


THE    THEORY   A^T5   PRACTICE    OF    rSTTERPOLATION. 


last  three  decimals  of  each  differ  only  allglMii  from  the  ([uantity 
0.00500,  or  half  a  unit  in  the  second  decimal  place;  and,  moreover, 
that  the  actual  difference  is,  excepting  the  first  function,  alternately  in 
excess  and  defect.  This  condition  will  rarely  obtain,  and  is  here  selected 
only  to  illustrate  the  limiting  case. 

If  now  we  drop  the  last  three  decimals  of  F,  we  obtain  a  series 
of  approximate  values,  denoted  by  -F.  The  following  table  gives  F, 
true  to  the  nearest  unit  of  the  second  decimal,  together  vpith  its 
differences : 


T 

F(T) 

J' 

J" 

jii, 

Jiv 

Jv 

Jvl 

0 
1 
2 

3 
4 

5 
6 

7 
8 
9 

8.43 
6.41 
5.89 
6.54 
7.66 
8.56 
8.65 
7.86 
6.76 
7.01 

-2.02 
-0.52 
+  0.65 
1.12 
0.90 
+  0.09 
-0.79 
—1.10 
+  0.25 

+  1.50 

1.17 

+  0.47 

-0.22 

0.81 

0.88 

-0.31 

+  1.35 

-0.33 
0.70 
0.69 
0.59 

-0.07 
+  0.57 
+  1.66 

-0.37 

+  0.01 

0.10 

0.52 

0.64 

+  1.09 

+  0.38 
0.09 
0.42 
0.12 

+  0.45 

-0.29 
+  0.33 
-0.30 
+  0.33 

(C) 


Table  (C)  corresponds  to  Table  (C)  of  §30.  It  will  be  observed 
that  r  and  J^',  in  (C),  represent  z/<"  and  J"+'^,  of  Table  (C).  The 
differencing  in  (C)  is  not  carried  beyond  J^' ,  because  of  the  alterna- 
tion of  -\-  and  —  terms. 

The  above  values  of  F  may  be  written  as  follows: 


F 

=    F   +   / 

8.43 

=  8.42511  +  0.00489 

6.41 

=  6.40508  +  0.00492 

5.89 

=  5.89492  -  0.00492 

The  quantities  in  the  last  colunni  therelbre  represent  the  residual 
terms  denoted  by  f  in  the  preceding  section.  Expressing  these  values 
in  units  of  the  second  decimal,  we  have  the  following  table  of  /  and 
its  differences: 


THE    THEORY   AND   PRACTICE    OF    INTERPOLATION. 


59 


T 

/ 

J' 

J" 

J'" 

Jiv 

jv 

J" 

0 

1 
2 

3 
4 
5 

6 

7 
8 
9 

+  0.489 
+  0.492 
-0.492 
+  0.492 
-0.492 
+  0.492 
-0.492 
+  0.497 
-0.481 
+  0.488 

+  0.003 
-0.984 
+  0.984 
-0.984 
+  0.984 
-0.984 
+  0.989 
-0.978 
+  0.969 

-0.987 
+  1.968 
-1.968 
+  1.968 
-1.968 
+  1.973 
-1.967 
+  1.947 

+  2.9.^5 
-3.936 
+  3.936 
-3.936 
+  3.941 
-3.940 
+  3.914 

-6.891 

+  7.872 
-7.872 
+  7.877 
-7.881 
+  7.8u4 

+  14.763 
-15.744 
+  15.749 
-15.758 

+  15.735 

-30.507 
+  31.493 
-31.507 
+  31.493 

(B") 


It  will  be  observed  that  the  quantities  of  Table  (B")  are  close 
approximations  to  the   (limiting)  valnes  given  in  Table  (B'),  of  §30. 

Let  ns  now  apply  JS^ewton's  Formula  to  interpolate  the  valne  of 
F  which  corresponds  to  T  =  0.40,  in  Table  (C) .  Neglecting  differ- 
ences beyond  J^,  we  take  from  Table  I  (for  n  =  OAO) ,  and  from 
Table  (C),  the  quantities  to  be  employed.     The  result  is  as  follows: 


J'o    = 

+  8.43 

A   =    +0.40 

a  =    -2.02 

Aa    = 

-0.8080 

B  =    -0.12 

b    =    +1.50 

Bb    = 

-0.1800 

C  =    +0.064 

e   =    -0.33 

Cc    = 

-0.0211 

D  =    -0.0416 

d  =    -0.37 

Dd  = 

+  0.0154 

JS  =   +0.02995 

e   =    +0.38 

Ee    = 

+  0.0114 

•  ••  K  = 

+  7.4477 

Whence,  we  write  for  the  value  of  the  interpolated  function, 


(K)   =  7.45 

=  7.44,77  +  0.00,23   =   i^„ +0.00,23 


(97) 


Computing  the  true  value  F„  from  (96),  we  obtain 


F.,  =  7.4320416  + 


(98) 


Hence  the  value    (i^„)  ^7.45,   interpolated  from  Table  (C),  is  in  error 
by  1.8  units  of  its  last  place. 

The  value  of  Q  is  the  result  obtained  by  interpolating  F^  from 
Table  (A'),  neglecting  differences  after  J^.  Thus  we  determine  Q  as 
follows : 


60  THE    THEORY   AXB   PRACTICE    OF   INTERPOLATION. 

/;  =  -1-8.4201 10 

A   =    +0.40  «o  =    -2.02003  Aa,  =  -0.S0S012 

B  =    -0.12  b,  =    +1.50987  />'/.„  =  -0.181184  + 

C  =    +0.0()4  c„  =    -0.3o'.)55  6V„  =  -0.023011  + 

D  =    -0.0416  do  =    -0.30109  /^(/„  =  +0.012525  + 

E  =    +0.02995  eo  =    +0.23237  A\  =  +0.006959  + 

.-.   Q  =  +7.432387  + 

The  value  of  7?g  is  computed  from  Table  (B")   in  the    same  man- 
ner that  Q  has  just  been  obtained  from  (A').     Thus  we  find 


/o    = 

+  0.489 

A   =    +0.40 

«„   =    +   0.003 

A'a„  = 

+  0.001 

B  =    -0.12 

/?„  =    -   0.987 

^A,  = 

+  0.118 

C   =    +0.064 

y„   =    +   2.955 

Cyo   = 

+  0.189 

D  =    -0.0416 

S,  =    -  6.891 

i)8o  = 

+  0.287 

E  =    +0.02995 

£„  =    +14.763 

&„    = 

+  0.442 

.-.  (In  units  of  the  second  decimal)  Ii\  =    +1.526  [Cf.  (83)] 

Now,  from  (91)  we  have 

K  =    (,>  +  B,  (99) 

Substituting  the  above  values  of  Q  and  Br,,  we  find 

F„  =  7.4324  +  0.01,53   =  7.4477 

which  agrees  with  the  result  obtained  directly  from  Table  (C). 

Since  the  sixth  differences  in  Table  (A')  are  constant,  it  follows 
that  the  true  value  F„  diffei-s  fi-om  the  above  value  of  Q  only  by  the 
term  in  J"  of  Newton's  Formula.  Now,  the  coefficient  of  J"  is  found 
from  Table  (D)  of  the  last  section  to  be  approximately  — 0.0230.  Hence, 
with  z/"  =  -j-O-OlSOT,  we  derive 


F„  =  Q-  (0.0230  X  0.01507) 

=  Q-   0.000346 

=  7.432387  -  0.000346 

=  7.432041 


(nearly) 


which  agrees  with   (98).     The  second  of  these  equations  gives 

Q  =   ^„  +  0.000340  + 

Substituting  this  value  of   Q  in   (99),  we  have 


F„  =  7^„  +  iis  +  0.0346 


THE    THEORY   AND   PRACTICE    OF   ENTERPOLATION. 


61 


where  tlie  numerical  term  is  now  expressed  in  llie  same  unit  as  R^. 
With  the  above  determined  value  of  ^5(^ -]-1.526),  the  last  equation 
becomes 

F^  =   F„  +  1.56 

Finally,  since  we  were  obliged  to  write  (i^„)  greater  than  F^  by 
0.23  units,  it  follows  that  the  actual  error  of  interpolation  in  this 
instance  is  1.56 -|- 0.23,  or  approximately  1.8  units  in  the  second  deci- 
mal place;    which  agrees  with  the  result  previously  obtained. 

32.  As  a  more  practical  application  of  Newton's  Formula,  we 
take  the  following 

Example.  —  Fi-om  the  appended  table,  find  the  sun's  right-ascension 
for  April  20'^  O''. 


Date 
1898 

Sun's  R.A. 

J' 

J" 

J'" 

J'v 

April    1 

6 

11 

16 

21 

26 

May     1 

6 

h       m       8 

0  43  20.30 

1  1  34.07 
1  19  52.99 
1  38  19.59 

1  56  55.84 

2  15  43.08 
2  34  42.36 
2  53  54.74 

III         8 

+  18  13.77 
18  18.92 
18  26.60 
18  36.25 
18  47.24 
18  59.28 

+  19  12.38 

S 

+   5.15 

7.68 

9.65 

10.99 

12.04 

+  13.10 

8 

+  2.53 
1.97 
1.34 
1.05 

+  1.06 

S 

-0.56 

0.63 

-0.29 

+  0.01 

Letting    t 


April  16,    we  have 


30-16 
5 


0.80 


Then,  from  Table  I,  and  the  above  differences,  we  find 


m      6 

F^      =       1  38  19.59 

A   =   +0.80 

a„  =    +18  36.25 

Aa„    =    +0  14  53.000 

B  =   -0.08 

io    =    +       10.99 

Bb^    =    -             0.879 

C  =    +0.032 

c„    =    +         1.05 

C<'„    =    +             0.034 

D  =    -0.0176 

d^  =    +         0.01 

Dd^  =                   0.000 

•.  Suu's  R.A.,  April  20'»  0" 


=       1  53  11.75 


which  is  the  value  given  in  the  American  Fjjhemeris  for  1898. 

33.     Since  the  value  of  n  in   the   preceding   example   is   only  0.2 
less  than  unity,  it  is   more   convenient   to   intei'polate   backivards  from 


02 


THE  THEORY  AND   PRACTICE  OF  INTERPOLATION. 


April  21,  by  means  of  (75).     Thus,  from    Table  I   (for  ?;  =  0.20),  and 
the  tabular  diiferences,  we  find 


n» 

8 

F,        =       1  56  55.84 

A  =    +0.20 

a_,  =    +18 

36.25 

-A(,_,   =    -0     3  43.250 

B  =    -0.08 

b^„    =    + 

9.65 

+  £i_„    =    -             0.772 

C  =    +0.048 

c_3    =    + 

1.97 

-Cc_3    =    -             0.095 

Z>  =    -0.0336 

d^   =    - 

0.56 

+  Dd_^  =    +             0.019 

.:  Sun's  E.A. 

,  Apri' 

20" 

O^             =       1  63  11.74 

which  agrees  within  0^01  of  the  first  result.  Whenever  a  check  is 
considered  necessary,  the  interj^olation  may  be  performed  by  both 
methods. 


Transformations  or  Newton's  Formula. 

34.  Modification  of  the  Foregoing  Notation  of  Differences:  Stir- 
llng's  Formula.  —  In  Newton's  Formula  of  interpolation  we  use 
differences  which  depend  only  upon  the  functions  F^,  F-^,  Fo,  .  .  .  .; 
the  functions  pi'cceding  F^,  whether  given  or  not,  are  in  no  way  in- 
volved. We  shall  now  transform  Newton's  Formula  in  such  a  man- 
ner as  to  involve  differences  both  preceding  and  following  the  function 
from  which  we  set  out.  The  resulting  formulae  will  in  general  be 
more  convenient,  rapidly  convergent,  and  accurate  than  Newton's 
Formula. 

In  the  schedule  below,  the  pi-eceding  notation  of  differences  is 
modified:  the  even  differences  which  fall  on  the  horizontal  line  through 
Ff,  are  now  denoted  by  the  subscript  zero,  as  ?>o  and  </„;  all  differences 
above  this  line  are  indicated  by  accents,  as  a,  V ,  c,"  etc.;  while  all 
differences  helom  the  horizontal  line  through  F^  ai-e  indicated  by  sub- 
scripts, as  ai,hi,  C.2,  etc.  The  new  schedule  of  differences  Avill  then 
be  as  follows: 


T 

FiT) 

J' 

J" 

J'" 

Jiv 

Jv 

t  -  2(0 

t—   01 

a" 

h' 

c" 

(/' 

e" 

t 

F, 

a' 

K 

<:' 

d. 

e' 

t+    0) 

F, 
F„ 

«i 

f>i 

''i 

di 

«! 

it  +  2a) 

a.. 

b„ 

<■-. 

(L 

e.. 

t  +  3u, 

Fs 

«3 

C3 

e. 

THE  THEORY  AND  PRACTICE  OF  INTERPOLATION.  63 

To  derive  Stirling's  Formula:    Applying  IN^ewton's  Formula  to 
the  above  schedule,  we  find  for  the  value  of  F„, 

F„  =   F^  +  na,  +  lU,,  +  Cc^  +  Dd^+  Ee,+   .    .    .    .  (100) 

where,  as  before,     B,  C,  D,  E,  .  .  .  .     represent  the  binomial  coefficients 
of    z/",  J'",  J'^,  J^,  .   .   .   .  ,     respectively.     Let  us  now  put 

a   =   H"'  +  «i)     '     «   =    i(«'  +  '"i)      '     e   =   +(e'+«i)  (101) 

from  which,  with  the  relations 

rti  —  a'   =   h^     ,     Ci  —  c'   ^   d^     ,     ?!    =    e'  +    .   .   . 

we  obtain 

flj   =   a  +  \b^     ,     c'  =.  c-\d^     ,     c,  =  c  +  it^,     ,     e,   =   e+   .  .  .  (102) 

Using   the   equations  (102),  together  with  the   relations  given  in  §23, 
we  find 


a^   =   a  +  ^h^ 


f'l    ~   *o  "*"  "i   =    *o  "''''"'"  "2"  '^o 

c„    =  c' +  2d^  + e,=  c+^d^+ e+     .    .    .       }  (103) 

d„  =  d„+'2e,+    .    .    .  =d„  +  2e+  .    .    . 

eg    =   Cj  +    .    .    .    =    e  +   .    .    . 

Upon    substituting   these   values   of    (h,  hi,  C2,     ....      in    (100),  the 
latter  becomes 

F^=::t\  +  n{a  +  U;)  +  D(J>^  +  c  +  kd,)  +  C{c  +  id^  +  e+  .    .)  +  D{d.^  +  2e+  .    .)  +  Ee+.    . 
=  F^  +  na  +  {B  +  ^)h^  +  {C+B)c  +  {D  +  lC+\B)da+{E+2D+C)e+  .    .    .    . 

Substituting  in  the  last  equation  the  values  of  B,  C,  D,  E,    namely, 

^   ^   n{n-V)  ^  ^   n{n-\)..{n-Z) 

11  '  li 

^  ^   n{ri-V){n-2)  ^  ^  n(n~l)  .  .  (n-4) 

li  '  11 

we  finally  obtain 

n-          w(a=-l)         reV«=-l),       w  (w=-l)(ra''-4)  ,,„,, 

F„  =  Fo+na+-^b,+  -^ '-0+      ^ ^^     X  +  -^ j^ 'e+   .    .  (104) 

which   is   known    as   Stirling's  Formula.      The  even   differences   em- 
ployed in  this  formula  are  those  falling  on  the  hoiizontal  line  through 


64 


THE    THEORY   AND   PRACTICE    OF   INTERPOLATION. 


i^oj  the  odd  differences  are  the  means  of  those  which  fall  immediately 
above  and  below  this  line,  as  defined  by  (101). 

Table  II  gives  the  values  of  Stirling's  coefficients  for  the  argu- 
ment n.  A  glance  at  this  table  shows  how  much  more  rapidly  these 
coefficients  converge  than  those  of  Newton's  Formula. 

Example.  —  From  the  table  below,  find  the  K.A.  of  the  sun  for 
April  20''  0". 


Date 

1898 

Sun's  R.A. 

J' 

J" 

J'" 

Jiv 

April    1 

6 

11 

h       m       8 

0  43  20.30 

1  1  34.07 
1  19  52.99 

m      8 

+  18  13.77 
18  18.92 
18  26.60 

S 

+   5.15 

7.68 

s 

+  2.63 
1.97 

s 

-0.50 

16 

1   38  19.59 

9.65 

0.63 

IS  36.25 

18  47.24 

18  59.28 

+  19  12.38 

1.34 

1.05 

+  1.06 

21 

26 

May      1 

6 

1  56  55.84 

2  15  43.08 
2  34  42.36 
2  53  54.74 

10.99 

12.04 

+  13.10 

-0.29 
+  0.01 

Taking   t  =  April  IG    (as  in  §32),  we  have 

n  =  52zl?  =  0.80 

The  horizontal  lines  drawn  in  the  body  of  the  table  indicate  the 
differences  to  be  employed  in  (104),  as  follows: 

(1)  The  required  values  of  Fq,  J",  and  J"  are  those  included 
between  two  lines; 

(2)  The  required  values  of  J'  and  J'"  are  the  means  of  the 
quantities  separated  hy  a  single  line. 

As  before,  we  shall  denote  the  coefficients  of  /I',  J",  J'",  ....  by 
A,  B,  C,  .  .  .  .  Taking  their  values  from  Table  II,  with  n  =  0.80, 
and  forming  the  required  differences  as  indicated,  we  obtain 


/;     =       1  38  19.59 

A   =    +0.80 

a    =    +18  31.425 

Aa     =    +      14  49.140 

i   =    +0.32 

i„   =    +         9.05 

Bb„    =   +             3.088 

C  =    -0.048 

c     =    +         1.00 

Cc     =    -             0.080 

D  =    -0.0096 

d„  =    -         0.03 

Dd^  =    +             0.006 

Sun's  R.A.,  April  20''  0" 


=       1  53  11.74 


which  agrees  exactly  with  the  result  foimd  in  §33. 


THE    THEORY    AND    PRACTICE    OF    INTERPOLATION. 


65 


8;").  Bdcl-irard  Tnterpolation  hy  Stirling's  Formula.  —  When  the 
forward  interval  approaches  unity,  it  will  be  more  convenient  t(»  pro- 
ceed hacliranl^  from  the  folloAviiig  function  by  the  formula 


„  n-  ,         «.  (rt^  — 1)         w-(7i^  — 1) 

/'_„=   F,  -  na  +  2  l>«  -     ^  g      '  <•  +  ^24        ' 


120 


e  + 


(105) 


the  coefficients  of  which  are  taken  from  Table  II  with  the  argument 
n,  as  before.  It  will  be  observed  that  (10;'))  is  derived  from  (104)  by 
merely  writing  — //  ibr  ??  in  the  latter;  or,  by  supposing  the  given 
series  to  be  inverted,  and  hence  (Theorem  III)  changing  the  signs  of 
a,  c,  and  <\ 

Example. — Solve  the  preceding  example  by  (lOo) ;  that  is,  find 
the  sun's  K.A.  for  April  20''  O**  by  backward  interpolation. 

Taking    t  =  April  21,    we  have 

M   =  ?lz^  =  0.20 

r> 

The  differences  are  formed  for  the  date  April  21  in  the  same  manner 
as  found  above  foi*  April  20;  thence,  taking  the  coefficients  from 
Table  II,  with    u  —  0.20,    we  find 


m 

J 

F,    =   1 

56  55.84 

A 

=  +0.20 

a  =  +18 

41.745 

—  Aa   =  — 

3  44.349 

B 

=  +0.02 

K    =  + 

10.99 

+  BK=    + 

0.220 

C 

=  -0.032 

c  =  + 

1.20 

-Cc    =    + 

0.038 

D 

=  -0.0016 

do   =  - 

0.29 

+  Dd,= 

0.000 

.-.  Sun's  R.A. 

,  April 

20'" 

0"      =   1 

53  11.75 

36.    Example.  —  Use  Stirling's  Formula  to  compute  log  sin  9°  22' 


fi'om  the  following  table: 


T 

Log  sin  T 

J' 

J" 

J"' 

Jiv 

Jv 

O 

6 

7 
8 

9.01923 

9.08589 
9.14356 

+  6666 
5767 

5077 

-899 
690 

+  209 
147 

-62 

+  17 

9 

9.19433 

543 

45 

4534 
4093 

+  3728 

102 
+  76 

+  19 

10 
11 
12 

9.23967 
9.28060 
9.31788 

441 
-365 

-26 

Here  we  have 


t  =  9° 


n  =  ".   ^  0.36667 

60 


G6 


THE    THEORY    AND    PRACTICE    OF   INTEKPOLATION. 


and  we  therefore  obtain 


/''„     = 

9.19433 

A   =    +0.3GGG7 

a    =    +4805.5 

Aa     =    + 

17G2.0 

£  =    +0.0G722 

/,,,  =    _   543 

Bl'o   =    - 

3G.5 

C  =    -0.05289 

'■    =    +   124.5 

Cr      =    - 

C.G 

/;  =   -0.00485 

d,  =    -     45 

Dd,  =    + 

0.2 

E  =    +0.01022 

e     =    +      18 

Ee     =    + 

0.2 

.-.  Log  sin  9° 

OO' 

= 

9.21152.3 

The  ti'ue  vahie  to  six  decimals  is  9.211526. 

37.  T/ic  Algehraic  Mean.  —  It  may  be  Avell  to  observe  that  in 
taking  the  mean  of  two  quantities  having  Hke  signs,  and  of  nearly 
the  same  magnitude,  it  is  easier  to  add  one-half  their  dijferefice  to  the 
lesser  number,  than  to  take  one-half  the  sum  of  the  two  quantities. 
That  is,  we  proceed  according  to  the  identity 

iix  +  i/)   =  a-  +  i(i/—x) 

in  which  we  suppose  y  numerically  greater  than  .c.     Thus,  in  the  last 
example,  instead  of  taking 

a  =   ^  («'  +  «,)   =  ^(5077  +  4534)   =   ^9611)   =    +4805.5 

it  is  easier  to  follow  the  equivalent  formula 

a   =   f,^-i(a,-a')    =   a,~h_K  =  4534  +  ^(543)   =    +4805.5 

Similarly,  we  find 

c  =   102  +  22.5  =    +124.5 

Per  contra,  to  form  the  mean  of  two  quantities  having  unlike 
signs,  and  differing  but  little  in  magnitude,  it  is  easier  to  take  their 
algebraic  sum  and  then  divide  by  two.     For  example,  given  the  values 


we 


find 


F(T) 

J' 

J" 

F, 
F, 

-4226 

+  5088 

+  9314 

a  =   i  (5088-4226)   =  i(  +  862)   =    +431 


With  these  precepts,  the  required  7nean  diffei'ences  of  interpolation  are 
very  I'cadily  taken. 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION.  67 

38.  Bessel's  Formula.  —  We  now  pass  from  Stirling's  Formula 
to  another,  somewhat  similar,  wherein  we  employ  the  odd  differences 
«!,  Ci,  e,,  which  fall  on  the  horizontal  line  between  Fo  and  Fi,  and 
the  means  of  the  even  differences  falling  immediately  above  and  below 
this  line.     Using  the  schedule  on  page  (32,  let  us  put 

Then,  since     ?>, — h„^c^,     and     d^  —  r/i,:^'",,     these  equations  give 

^'o  =  i-iq         ,         (1,  =  d-^_e,  (107) 

Let  us  write  the  formula  (104),  for  brevity, 

i.'„   =   F^+na+fb,+  Cc+Dd,+  EeJr    ....  (108) 


where 


C   =  g ,  i>  -   24  '  ^'  -  120  ^     ^ 

Now,  by  means  of  (102)   and  (107),  we  derive 

a     =   ffj  — 4-io   =    fli— K/-  — ^f,)    =    «!  — i^'  +  i^i 

*o  =   f>  —  i  ''i 

c  =   c,  -id,   =   c,-i(d-ie,)   =   c,-id  +  ie,       }  (110) 

fZ„  =   d  —ws^ 

e  =  «!  —  .    . 

Upon   substituting  these  values   of    a,  h„,  c,  .  .  .  .    in   (108),    we  have 

F„  =  F,  +  7i{a,-ib  +  ir,)  +  f{b-ic,)  +  C(c,-id  +  ie,)+I){d-ie,)  +  E(e,~  .  .)  +   ■  •  • 

Finally,  substituting  in  the  last  equation    the  values  of    (J,  D,  E, 
from   (109),  we  obtain 

„           „                 7i(7j  — 1),       n(n  —  l)(n  —  i) 
F,   =   F„+7w,+     ^2      ^+— i^ ^'■i 

(>.+  l)n(n-l)(M-2)          (n  +  l)<n-l)(»-2)(n-j-) 
+  24 + 120 ^~^   '    ■    ■  ^     ^ 

which  is  Bessel's  Formula  of  interpolation,  commonly  regarded  as 
the  most  convenient  and  accurate  of  the  several  forms  in  use.  The 
odd  differences  here  employed  are  those  which  fall  on  the  horizontal 
hue  between  Fq  and  F-^^,  as  shown  in  the  schedule  on  page  62;  the 
even  differences  are  the  means  of  those  falling  immediately  above  and 
below  this  line,  as  defined  by   (106). 


68 


THE    THEORY   AXD   PRACTICE    OF    INTERPOLATION. 


Table  III  gives  Bessel's  coefficients  for  the  argument  n. 
Example.  —  Use  Bessel's  Fonnnla  to  compute   log  sin  9°  22'  from 
the  table  below: 


T 

Log  sin  T 

J' 

J" 

J'" 

Jiv 

Jv 

O 

6 

7 
8 

9.01923 
9.08589 
9.14356 

+  6666 
5767 

5077 

-899 
690 
543 

+  209 
147 

-62 

+  17 

9 

9.19433 

45 

4534 

102 

+  19 

10 
11 
12 

9.23967 
9.28060 
9.31788 

441 

-26 

4093 
+  3728 

-365 

+  76 

We  have,  as  in  §36, 


9° 


11  =  ()..")6667 


The  horizontal  lines  drawn  in  the  table  indicate  that  the  values 
of  F^,  J',  J'"  and  J",  to  be  employed  in  (HI),  are  those  included 
between  the  parallel  lines;  while  the  required  values  of  /J"  and  j"  are 
the  means  of  the  quantities  separated  by  the  single  line.  Foiming  the 
differences  thus  indicated  and  taking  their  coefficients  from  Table  III, 
with     n  =  0.36  | ,     we  obtain 


^0 

= 

9.19433 

A    =  +0.36667 

«, 

=  +4534 

Aay 

= 

+ 

1662.5 

B   =  -0.11611 

/, 

=  -  492 

Bh 

= 

+ 

57.1 

C   =  +0.00516 

C: 

=  +  102 

Cc, 

= 

+ 

0.5 

n  =    +0.02160 

d 

=  -  36 

Dd 

= 

— 

0.8 

E   =  -0.00057 

«i 

=  +  19 

Ee^ 

= 

0.0 

.  Lost  sin  9° 

22' 

= 

9 

21152.3 

which  agrees  exactly  with  the  value  found  in  §36. 

39.     Example. —  Find  by  Bessel's  Formula  the  value  of  10''  from 
the  following  table  of  T^. 


T 

Ti 

J' 

J" 

J'" 

Jlv 

jv 

-  8 

-  3 
+  2 

+  4096 
81 
16 

-  4015 

-  65 
+  2385 

+  3950 

2450 

15950 

-  1500 
+ 13500 

+  15000 
15000 

0 

7 

L'401 

18335 

28500 

(1 

12 
17 

+  22 

20736 

83521 

+  234256 

44150 

+  87950 

+  15000 

62785 
+  150735 

+  43500 

THE  THEORY  AND  PKACTICE  OF  INTERPOLATION.  69 


Taking     f  =z  7,     we  have 
Therefore  wc  find 


n   =   ?t!   =   O.fiO 


F,     =    +   2401 


A   =    +0.60                 «i   =    +18335  Aa,  =  +11001 

5  =    -0.120               l>     =    +30200  Bb  =  -  3624 

C  =    -0.0040             i\    =    +28500  C\  =  -     114 

Z>  =    +0.0224             d    =    +15000  IM  =  +     336 

.-.  lO-"  =  +10000 

40.  Bacl-ward  Interpolation  Ixj  Bessel's  Formula.  —  To  find 
i^_„  by  Bessel's  Formula,  we  conceive  the  series  given  on  page 
()2  to  be  inverted;  the  required  function  is  then  found  by  interpo- 
lating toward  from  Fi,  toward  i^_i  with  the  interval  n.  Hence,  the 
differences  to  be  used  in   (111)   are  — 

We  therefore  have 

F_..  =   /;-  na'+!^  .  ?4'-'  -  "(-^>^-^)  .'+....  (111«) 

the    coefficients,    as    in    (111),    being    taken    from    Table  III    Avith    the 
argument  n. 

Example.  —  Find  10*  from  the  table  of  §39,  by  means  of  (111^/). 

Taking     t  =  12,     we  find 

,,   =   -til'  =   0.40 

The  differences    are    here    the    same    as    in   the  last   example;    thus    we 

obtain 

I\  =    +20736 

A   =    +0.40       a'  =  +18335      -Aa'  =  -  7334 

B   =  -0.120      y^^'  =  +.30200      +B.'^^^   =    -  3624 
C  =    +0.0040     c'  =   +28500      -Cc'  =    -     114 

D=   +0.0224     —^  =   +15000     +7;.'i»l^'=  +  336 

.-.  10^  =  +10000 

41.  Propertij  of  Bessel's  Coefficients.  —  If  we  take  from  Table  III 
the  coefficients  for  /I",  J'",  /]",  z/^,  with  the  argument  n  =^  0.30,  and 
also  with     ji  =  0.70  (=:  1.00  —  0.30),     we  find  the  folloAving  values: 

n  B  C  BE 

0.30     -.10500     +.00700     +.01934     -.00077 
0.70     -.10500     -.00700     +.01934     +.00077 


70  THE  THEORY  AXD  PKACTICE  OF  INTERPOLATION. 

It  will  l)c  observed  that  the  coefficients  are  here  numerically  the  same 
for  the  arguments  //  and  1  —  n  ;  having  like  signs  for  the  even  orders, 
and  opposite  signs  for  the  odd  orders  of  differences. 

More  generally,  let  us  denote  the  values  of  Bessel's  coefficients 
for  J",  J'",  /)'",  /T,  ....  taken  with  the  argument  n,  by  B,  C,  D,  E,  . .  . ., 
respectively;  and  the  corresponding  values  taken  with  the  ai-gument 
1  —  n  by  Bi,  Ci,  Di,  E^,  .  .  .  .  An  inspection  of  Table  III  then 
shows  that  we  have 

B,  ==  +B  \ 

C,  =   -C  ) 

I),  =    +I>  )  (112) 


To  establish  these  relations  generally,  we  write   (111)   in  the  form 

F„  =   /•;  +  »»i  +  i»  +  C(\  +  Dd  +  i>,  +    .    .    .    .  (113) 

Now,  the  value  of  F^  may  also  be  obtained  by  interpolating  hacl'- 
loardfy  from  Fi  with  the  interval  1  —  // ;  the  diffi^rences  thus  involved 
will  I)e  exactly  the  same  as  in  (113).  Hence,  after  the  manner  of 
formula  (Ilia),  we  have 

F„  =  F,-  (1-m) «!  +  BJ>  -  C\c,  +  D,d  -  E^e^+  .    .    .    .  (114) 

But  we  have,  also, 

^1  -  (i-'0«i  =   (^i-''i)  +  "«i  =   ^0  +  ""l 

Whence,   (Hi)   becomes 

F,   =    7';  +  na,  +  B,h  -  C,'\  +  l\d  -  /iV,  +  .    .    .    .  (115) 

which,  subtracted  from  (113),  gives 

0   =   {B-B{)l,+  {C+C,)c^+{D-D,)d+.    .    .    .  (116) 

The  equation  (116)  is  true  in  all  cases  to  which  the  formulae  of 
interpolation  are  applicable;  it  is  therefore  true  when  F(T)  is  a 
rational  integral  function  of  the  second  degree.  But,  in  the  latter 
case,  the  second  differences  being  constant,  we  have 

c^  =  d  =  e^  =  .    .    .    .    =  0 
The  equation  (116)  tlien  becomes 

0   =   (B-B,)b 


THE  THEOEY  AND  PEACTICE  OF  INTERPOLATION.  71 

Hence,  since  h  cannot  vanish,  we  have 

B^  =   +B 
This  result  i-ecluces  (11 G)  to  the  form 

0   =   (C+Ci)ci  +  (Z)-7>i)t/+ (A''+/i\)c,  +  ....  (117) 

Again,  we  may  suppose  J'"  constant;    tliat  is,  we  may  put 

rf      =      ?!      =      ....=      0 

The  equation   (117)   then  becomes 

0   =   {C+C\)c, 
or 

C^=   -C 

By  repeated  application  of  this  reasoning,  we  prove  that  the  rela- 
tions (112)   are  true  generally. 

It  follows  that  the  numerical  process  involved  in  finding  F„  by 
Bessel's  Formula  is  identical  whether  we  interpolate  forward  from  F^^ 
or  backward  from  F^,  except  for  the  terms  in  F  and  J'.  Hence  little 
or  no  check  is  afforded  by  pei'forming  the  interpolation  by  both  methods. 
When  such  a  check  is  deemed  necessary,  Bessel's  and  Stirling's 
Formulae  should  both  be  used. 

42.  Relative  Advayitages  of  Newton's,  Stipo^ing's,  and  Bessel's 
Formtdae.  —  In  practice,  the  only  important  application  of  Newton's 
Formula  consists  in  interpolating  functional  values  near  the  heginninr/ 
or  end  of  a  given  series.  The  selection  of  this  formula  is  then  a 
matter  of  necessity  rather  than  of  preference. 

In  all  other  cases,  either  of  the  more  rapidly  converging  formulae 
of  Stirling  or  Bessel  should  be  employed.  Regarding  a  choice 
between  these  two,  when  Tables  II  and  III  are  available  there  would 
appear  to  be  very  little  advantage  one  way  or  the  other.  The  form 
given  by  Bessel  is  more  commonly  used,  and  is  perhajis  a  trifle  moi-e 
accurate  in  practice  than  Stirling's  form,  particularly  for  values  of 
n  in  the  neighborhood  of  one-half.  When  n  is  quite  small,  howevei", 
Stirling's  Formula  will  probably  be  found  more  convenient. 


72 


THE    THEOKY    ^^'^D   PRACTICE    OF    INTERPOL ATIOX. 


iSn[)pose  we  have  given  a  limited  table  of  functions,  as  follow; 


F(T) 

J' 

J" 

J'" 

Jiv 

F  „ 
F., 

a" 
a' 

"3 

'■1 

''0 

Assuming  that  fourth  differences  must  be  taken  into  account,  and 
that  fifth  differences  ai'e  to  be  neglected,  the  value  of  i^„  should  in 
this  case  be  comjiuted  by  Bessel's  Formula,  which  employs  the  mean 
of  the  quantities  (/„  and  d^ .  If,  however,  the  function  i^g  were  not 
included  in  this  series,  then  the  term  fZ,  would  not  be  given,  and  we 
should  proceed  b}^  Stirling's  Formula,  which  involves  d^  directly. 

Bessel's  Formula  is  particularly  simple  and  convenient  when 
?i  =  ^  ,  that  is,  Avhen  it  is  required  to  find  the  function  which  falls 
midway  between  F^  and  i'^, ;  this  important  case  will  be  fully  con- 
sidered in  a  later  section. 

43.  Simple  Interpolation. — When  frequent  interpolation  is  required, 
as  in  tables  of  logarithms,  trigonometric  functions,  etc.,  the  interval  of 
the  argument  is  usually  chosen  sufficiently  small  that  the  effect  of 
second  diffei-ences  may  be  neglected.  Bessel's  Formula  gives  in  this 
case 

F„   =   F,  +  na,  (118) 


To  interpolate  hack  wards  from  i^„,  that  is,  to  find  i^_„,  we  obtain 
from   (lllr/),  by  neglecting  second  and  higher  difl'crences. 


=   F,-  na' 


(119) 


Upon  these  formulae  the  process  of  simple  interpolafion  is  based. 
The  first  difference  to  be  used  in  either  case  is  the  value  falling 
between  Ff,  and  the  function  towaid  which   the  interpolation  proceeds. 

Frequently,  where  great  accuracy  is  not  required,  it  is  sufficient 
to  obtain  F„  by  simple  interpolation  even  when  the  second  diffi3rences 
are  considerable.     In  such  a  case,  suj^posing  that  the  third  diiferences 


THE    TIIEOKY   AND   PRACTICE    OF   INTERPOLATION.  73 

are   insensible,  we    observe  from    Bessei/s  Formula    that    the    error    of 
the  approximate  value  of  _F„  will  be  — 

8^„   =  !i(!^.i"  (120) 

The  maximum  value  of  — ^ — '  which  obtains  for  //^|,  is — |-; 
whence  we  have  the  following  result : 

]Vheii  second  differences  are  seiisildi/  coustaat,  the  inajciiuuiit  error 
of  //( net  ions  obtained  I)y  simple  interpolation  is   i  J" . 

Thus,  in  Tables  I,  II,  and  III,  the  values  of  the  coefHcients  for 
J"  (designated  above  as  B)  can  nevei'  be  in  error  by  more  than  I  of 
10  units,  or  1.2  units  in  the  fifth  decimal,  when  found  by  simple 
interpolation. 

44.  Tnterpolatloa  Livolving  Second  Differences,  by  Means  of  a 
Corrected  Fii-st  Difference.  —  When  the  second  differences  are  con- 
stant, or  nearly  so,  but  too  large  to  neglect,  their  effect  may  be 
included  (and  hence  an  accurate  value  of  F„  obtained)  by  the  follow- 
ing simple  method  : 

Since  third  differences  are  supposed  insensible,  Bessel's  Formula 
becomes 

T1  T,  "     ('1  —  1)     7 

which  may  be  written  in  the  form 


^,.   =    F,+  n 


1-n 


(121) 


Now,  because  third  differences  are  negligible,  we  may  write  ^o  foi-  h  in 
(121) ;    then,  putting 


V  2  y''» 

we  have  \    -   /  ^  (-^22) 

F„  =   F,  +  na, 

The  value  of  F^  is  thus  obtained  almost  as  readily  as  in  simple 
interpolation.  In  forming  the  quantity  -~  (which  is  simply  one-half 
the  complement  of  n  with  respect  to  unity),  only  an  approximate 
value  of  71  is  ordinarily  required.     The  value  of   a, ,    the  corrected  first 


74 


THE    THEORY    AXD    PRACTICE    OF   INTERPOLATION. 


(Uffcrence,  is  tluis  found  by  an  easy  mental   process   amounting  almost 
to  nirri'  inspection. 

Example.  —  Find  (8.2)^  from  the  following  values  of  T'^ : 


T 

7-2 

J' 

J" 

4 

7 

10 

13 

16 

49 

100 

169 

+  33 

51 

+  69 

+  18 
+  18 

Here  we  have 

t  ^  1         ?i   =   0.4         F^  =  49         a,  =   51         /'„  =   18 
Hence,  by  (122),  we  find 

a,   =   51  -  (0.3  X  18)       =   45.6 
.-.  F„  =  49  +  (0.4  X  45.6)   =   67.24 

This   result    is    exact,   because    the    second   differences    are    rigorously 
constant. 

45.  BacliiKird  Tnterpolation  hy  Means  of  a  Corrected  First 
Deference.  —  From  (111«),  neglecting  differences  beyond  J",  we 
obtain 


F_.,  =   J^„-/m'+-^ 


(«-l)      ft„  +  i' 


F.-na'Jr 


,      n{n-l) 


or 


F_..  =   F 


n  *  0 


Hence,  if  we  put 
we  have 


«'   =    a'  + 


l-» 


(123) 


(124) 


Example.  —  From  Hill's  Tables  of  Saturn,  the  following  pertur- 
bations ai-e  taken;  find  the  value  corresponding  to  the  argument 
T  =  30G82.38. 


r 

F(T) 

A' 

J" 

28800 
29760 
30720 
31680 
32640 

12.5751 
12.1998 
11.8315 
11.4700 
11.1148 

-3753 
3683 
3615 

-3552 

+  70 

68 

+  63 

THE  THKORY  AND  PKACTICE  OF  TNTEKPOLATION.  75 

Taking  t  =  30720,  we  have 


/<;   =   11.8315  n  =    '^^'(j^Q^"^'^   =  0.03919  (backward  Irom  F^) 

T    =  30G82.38  «'   =    -3G83 

o.    =  960  /;,,   =    +      68 


Usiiif?  0.04  as  a  sufficiently  accurate  value  of  //  in  determining  a', 
we  find  by  (124), 

LZ!!  =  1^2!   =  0.48 

«'   =    -3683  +  (0.48  X  68)   =    -3650 
.-.  F_,,  =   11.8315  -  [0.03919  X  (-3650)]   =   11.8458 

In  the  present  example  the  algebraic  signs  of  the  several  quanti- 
ties of  (124)  have  each  been  considered.  Now  it  is  important  to 
remark  that  in  the  majority  of  cases  no  attention  need  be  given  to 
these  signs;  for  in  this  fact  lies  the  chief  practical  advantage  of  the 
method.  Thus,  in  the  present  example,  we  are  interpolating  from  the 
third  function  toward  the  second;  the  value  of  J'  to  be  corrected  is 
the  difference  of  these  two  functions,  or  3683;  the  sign  we  disregard. 
The  correction  to  be  applied  to  this  number  is  0.48  X  68,  or  33.  Again 
neglecting  signs,  we  simply  apply  this  quantity  to  3683  in  such  a 
manner  as  to  obtain  a  result  falling  somcAvhere  between  the  numbers 
3683  and  3615  of  the  column  j'.  Hence,  we  decrease  3683  by  33, 
thus  obtaining  3650  for  our  corrected  first  difference,  a.  Finally, 
no!  =  143,  by  which  amount  we  increase  the  function  11.8315  (giving 
11.8458),  since  we  observe  that  the  functions  are  increasing  in  the 
direction  of  the  interpolation. 

A  partial  exception  to  this  mechanical  method  of  procedure  is  to 
be  observed  when  a^  and  a  have  opposite  signs ;  that  is,  when  //' 
changes  sign  in  passing  the  function  Fq.  In  this  case  the  sign  of  a 
must  be  noted;    we  then  have,  as  in  (122)   and  (124), 


re 


THE  THEORY  AXD  PRACTICE  OF  INTERPOLATION. 


For  exaiiiplo,  given  the  values  below  : 


r 

F{T) 

J' 

J" 

10 
15 
20 
25 

138 
538 
638 
438 

+  400 
+  100 
-200 

-300 
-300 

Suppose  it  is  required  to  find  F,  for  T  =  19.  We  let  t  =  20, 
F^,  =  038,  and  interpolate  hacl-wards  with  n  =  0.20.  To  obtain  a, 
decrease  100  by    O.lXIJOO,   or  120;     whence    a' —  — 20,    and  thei-efore 


lu'   =   638  -  [0.2  X  (-20)] 


642 


We  remark  in  passing  that  the  value  of  the  corrected  first  differ- 
ence, either  in  forward  or  backward  intei'polation,  is  always  contained 
between  the  limits    a^    and    <i' . 

The  number  of  instances  in  ])ractice  where  the  differences  beyond 
/"  may  be  neglected  is  ver}'  lai'ge.  The  precepts  given  above  are 
therefore  important,  and  should  be  practiced  by  the  student  until  theii- 
application  becomes  rapid  and  mechanical. 

46.  Correctioii  of  Erroneous  Functions  Ihj  Direct  Tiiterpolatioii  of 
the  Values  in  Question. — When  an  eri-or  has  been  detected  in  some 
one  function  of  a  series  by  the  method  of  differences,  as  explained  in 
§8,  it  is  often  possible  to  find  the  true  value  of  that  quantity  by 
direct  interpolation.  To  accomplish  this,  we  have  only  to  omit  from 
the  given  series  every  alternate  function,  the  incorrect  value  being  one 
of  the  number  rejected.  We  have  then  to  make  but  one  interpolation, 
midway  hetwee/n  two  functions  of  the  new  series,  to  obtain  the  value 
I'equired.  It  is  necessary,  however,  that  the  given  series  shall  include 
a  sufficient  number  of  functions  to  furnisli  an  adequate  schedule  of 
differences  in  the  abridged  table ;  furthei'more,  the  interval  of  the 
original  table  must  be  sufficiently  small  that  the  magnified  differences 
of  the  a])ridged  tal)!e  will  not  be  so  large  as  to  render  interjjolation 
impossible. 

AVe  illustrate  by  means  of  Examjile  III,  §9.  The  value  of  /8  for 
May  11.0  was  found  to  be  incorrect ;  hence,  to  find  the  true  value, 
we  omit  from  the  given  series  the   positions  for  eveiy  noon,  retaining 


TllK    THEORY    AND    PRACTICE    OF    INTERPOLATION. 


77 


only  the    values    for    each    midnight.       Thus    we    obtain    the    following 


ahi-idged  series  : 


Date 
1898 

^ 

J' 

J" 

J/" 

Jlv 

May  8.5 

9.5 

10.5 

11.5 

12.5 

-1  59  .54.2 
-0  44  27.0 
+  0  32  39.9 
1  46  12.4 
+  2  51  51.2 

0           /            II 

+  1  15  27.2 
1  17  6.9 
1  13  32.5 

4  1     5  38.8 

/              // 

4-1  39.7 
-3  34.4 

-7  .53.7 

-5  14.1 
-4  19.3 

+  54.8 

The    value    of    /3    for  May  11.0    i,s    now    readily    found    by    interpola- 
tion ;    for  this  purpose,  we  take 


t  =   May  10.5 


F,  =    4-0°  32'  .39".9 


n   =   0.50 


Since  but  one  value  of   z/*^    is  given,  namely     d„  r=.  5-1.8,     we  pro- 
ceed by  Stirling's  Formula  (see  §42);    thus  we  find 


O            /            11 

^0          = 

+  0°32'39"9 

A  =    +^ 

«    =    4-1  15  19.7 

Aa    = 

-1-0  37  39.85 

B  =    +1 

/,^    =    _       3  34.4 

Bb,   = 

-           26.80 

c  =  -A 

c     =    -       4  46.7 

Cc     = 

+           17.92 

D  =    -0.00781 

rf„  =    +           54.8 

Dd,  = 

-             0.43 

;8  (May  11.0,  1898)   =    4-1  10  10.44 


The  value  found  in  §9  by  the  method  of  differences  is  -(-1°  10'  10".6. 
The  result  just  obtained  by  interpolation  is  uncertain  within  nai-row 
limits,  because  we  have  no  knowledge  of  the  value  of  J"  in  the  above 
table.  The  value  1°  10'  10".6  should  therefore  l)e  taken  as  the  more 
probable. 

Had  the  value  of  y8  for  May  13..5  been  included  in  tlie  oi'iginal 
series,  our  abridged  table  would  have  yielded  two  values  of  J"  and 
one  of  j^'.  We  should  then  have  used  Bessel's  Formula  (see  §42)  to 
compute  the  latitude  for  May  11.0.  ISTow,  the  moon's  latitude  for  May 
18.5,  1898,  is  +3°  46'  22".2 ;  including  this  value  with  the  others  above, 
and  applying  Bessel's  Formula,  we  find     /S  =  +1°  10'  10" .57. 

47.  When  a  series  contains  several  incorrect  functions,  separated 
from    each    otlier    by   even   multiples    of  the   interval    w,    the    foregoing 


78  THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 

method   at  once  serves  for   the  determination  of  the   several  vahies  in 
qtiestion.     Thus,  in  the  series 

F      F     F      F      F 

let  lis  suppose  that  F^,  i'^a,  and  F-  are  in  error.     Then,  if  we  tabulate 
and  difllerenee  the  series 


F    F    F    F     F 

-'OJ     -'2J     -*4J     -'OJ      *K) 


the  required  values  are  easily  found  by  interpolation. 

Again,  when  two  adjacent  functions,  say  F^  and  F^,,  require  cor- 
rection, we  may  proceed  by  tabulating  every  tJiird  function  of  the 
given  series;    thns  we  obtain  the  abridged  series 

F      F      F      F 

from  which  the  values  of  Fi  and  Fr^  are  found  by  interpolating  with 
w  ^  I  and  I,  respectively.  Otherwise,  if  the  differences  of  the  latter 
sei'ies  are  too  large  for  accurate  interpolation,  we  may  omit  from  the 
original  table  every  alternate  function  only,  as  in  §4G.  The  resulting 
series, 

F      F      F      F      F 

will  therefore  contain  but  one  incorrect  value,  namely  Fi.  The  cor- 
rection to  Fi  may  then  be  found  by  the  method  of  differences,  whereas 
this  method  might  be  impracticable  if  applied  to  F^  and  jPj  simulta- 
neously.    Similarly,  we  may  correct  F^  by  the  differences  of 

F    F    F    F    F 
01-,  by  interpolation  from  the  corrected  series 

-^(1)     -^2.     Fi,     K,     J^'\>       ■      ■      ■      ■ 


Systematic  Interpolation — Subdivision  of  Tables. 

48.  Thus  far  we  have  considered  interpolation  as  a  process  I'or 
comi^uting  the  values  of  functions  for  occasional  oi-  .sper/a/  values  of 
the  argument,  simply.     We  shall  now  consider  the  subject  in  a  broader 


THE    TIIEOKY    AND    PRACTICE    OF    INTERPOLATION.  79 

sense,  and  find  that  interpolation  is  of  great  importance  as  applied 
in  a  more  extended  and  systematic  manner. 

When  a  complicated  function  is  to  be  computed  and  tabulated  for 
a  lai-o'e  number  of  equidistant  v-alues  of  the  argument,  oi-  when  the 
tabulai"  quantities  result  from  a  long  and  laborious  calculation,  it  will 
be  much  shorter  and  easier  to  make  the  direct  computation  foi-  a  less 
frequent  interval  than  is  finally  required,  and  thence  to  obtain  the 
intermediate  values  by  systematic  interpolation.     Foi-  example,  suppose 

the  function 

F{T)   =   700".4.3  sin2r-l".19  sin4T 

is  to  be  tabulated  for  every  10'  from  30°  to  G0° ;  we  should  begin  by 
computing  F(^T)  for  every  4th  degree  of  T.  Thus  we  should  obtain 
the  values  of  F{T)  for   T  = 

22°,  26°,  30°,  34°,  .    .    .    .  70°  ; 

the  calculation  being  extended  somewhat  beyond  the  assigned  limits 
in  order  to  facilitate  the  interpolation  which  follows.  These  quantities 
having  been  differenced,  and  corrected  for  accidental  erroi-s  if  neces- 
sary, the  middle  terms  are  then  found  by  interpolation  to  lialves.  We 
thus  obtain  the  series  F(^T)  corresponding  to   T  = 

26°,  28°,  30°,  32°,  ....  64° 

Interpolating  again  to  halves,  we  have  a  table  of  F {^T)  for  every 
degree  of  T.  A  thii'd  interpolation  to  halves  gives  the  function  for 
every  30'.  Finally,  interpolating  the  latter  series  to  thirds,  we  obtain 
the  required  table,  giving  F(T)  for  every  10'  of  the  argument  T. 
It  is  obvious  that  the  labor  of  computation  decreases  rapidly  Avith 
each  successive  interpolation. 

All  of  the  extended  tables  in  common  use,  such  as  tables  of  loga- 
rithms, sines,  tangents,  etc.,  have  been  subdivided  in  this  manner,  at  a 
saving  of  labor  almost  beyond  estimation.  In  fact,  interpolation  has 
undoubtedly  done  more  for  mathematical  science  than  any  other  dis- 
covery, excepting  that  of  logarithms. 

The  following  sections  will  be  devoted  to  the  derivation  of  formulae 
and  precepts  which  will  simplify  the  process  of  systematic  intei'polation 


DR.  F.  McEWEN 

80  THE    THEOm'   AND    PRACTICE    OF   INTERPOLATION. 

just  describL'd.  Instead  of  performing  a  separate  and  distinct  calcu- 
lation for  each  interpolated  function,  we  shall  develop  a  method  by 
which  the  required  values  are  obtained  by  .successive  (tdditiovs  of  the 
computed  differences  of  those  values. 

The  most  convenient  interpolation  to  perform,  either  in  an  isolated 
case,  or  as  applied  to  the  subdivision  of  an  extended  series,  is  interpo- 
lation to  /i((lres,  which  gives  the  function  corresponding  to  the  ^iieaii 
of  two  consecutive  tabular  values  of  the  argument.  This  case  will 
now  be  considered. 

40.  Interpolation  to  Halves.  —  If,  in  Bessel's  Formula  (111),  we 
put     y/  =  i,      the  coefficients  of  j'"  and   J"  vanish,  and  we  get 

^  =  ^;  +  i«i-i^  +  Tis'^-  •  •  •  ■  (125) 

Since     F^ — F^^^a^,     we  have 


Also,  by   (100),  we  have 


F,  +  ^a.,   = 


,^h±k 


d  = 


d,  +  d. 


Hence,  (125)  may  be  written  in  the  form 

.;  =  5di  _  J  (^±^)  +  ,,.C.±i?. ,_....  (126) 

which  is  the  foi'mula  for  interpolation  to  halves,  true  to  fiflh  differences 
inclusive.  The  differences  are  to  be  taken  according  to  the  schedule 
on  page  02. 

Supposing  that  fourth  differences  are  so    small    as    to    produce  no 
sensible  effect,  we  obtain  from   (120)   the  very  simple  formula 


true    to    third  diflerences    inclusive.     Hence,  to    interpolate    a    function 
midtvay  between  two  consecutive  tabular  values,  we  have  the  following 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


81 


Rule  :  Fro»i  f/te  uicdn  of  fJie  fino  (/irrji  fiDirtioxf^,  siihfracf  one- 
eiylifli  the  mean  of  the  second  differences  which  stand  opposite.  Tlu' 
result  is  true  to  third  differences  inclusive.  To  obtain  the  value  true 
to  fifth  differences  inclusive,  add  to  the  almve  residt  j|^  of  the  mean 
of  the  corresponding  fourth   differences. 

50.  Precepts  for  Systematic  Interpolation  to  Halves. — The  fore- 
going rule  applies  either  to  the  interpolation  of  a  single  function  into 
the  middle,  or  to  that  of  an  entire  series  of  values.  For  the  latter 
purpose,  however,  the  work  may  be  arranged  in  a  more  expeditious 
manner,  as  follows: 

For  convenience,  we  assume  for  the  present  that  4th  diflerences 
may  be  neglected;    accordingly,  if  we  put 


Sn'    =    t\ 


W  =  ^1  -  J'i 


F,  -  F, 


8^  =  F,-F, 


(128) 


we  obtain  from   (125), 


So'  =  i«i  - 
V  =  ia,- 
8,'  =  A-ttj  — 


2 
h,  +  L- 


K  +  k, 


(129) 


The  quantities  S'  defined   by  (128)   are   evidently  the   first   differ- 
ences,oi  the  interjjolated  series  ;  the  alternate  terms,  S,/^  ^ly  §4 ?   are 

computed  by  (129)   from  the  fii'st  and  second  differences  of  the  given' 
series  of  functions  ;    the  values  of     S/,  83',  Sg',  ....      are  not  computed. 
The  method  and  arrangement  of   the  work  are    shown  in  the  schedule 
below  : 


T 

F(T) 

8' 

8" 

a 

(3 

J' 

J" 

J'" 

t 

t  +  a, 
«  +  §<« 

t  +  lio 

F-. 

Fo 

F. 

F, 

Fz 

80' 

8/ 

83' 
84' 
85' 

80" 
81" 
8." 
8s" 
8," 
8." 

-<*•:*■) 
-<^) 
-'.(^) 

a' 

"1 
"2 
a.) 

''0 

82  THE    THEORY   AND    PIt;VCTICE    OF   USTTERPOLATION. 

The  differences  of  the  given  series  are  placed  in  the  last  three 
columns,  under  J',  //",  and  J'".  The  column  a  is  then  filled  in  by 
writing  opposite  each  of  the  quantities  ji  one-half  its  value.  The 
column  ^  is  also  computed,  each  term  being  mhius  one-eighth  the  mean 
of  the  tAvo  values  of  J"  which  stand  opposite.  The  alternate  quantities 
of  column  8'  are  then  found,  as  in  (129),  by  taking  the  sums  of  the 
con-esponding  terms  in  a  and  fi ;  the  results  are  written  immediately 
above  the  line  of  the  latter  terms,  so  as  to  fall  between  F^  and  jP,  ,  F^ 
and  F, ,  etc.,  i*espectively. 

Finally,  since  b}'  (128)   we  have 

F,^  =  F^  +  S;     ,     F,  =  h\  +  SJ     ,     F^  =  F,  +  8,'     ,      ....  (130) 

it  is  only  necessary  to  add  each  computed  value  of  8'  to  the  function 
immediately  preceding,  to  obtain  the  required  middle  functions.  Hav- 
ing thus  completed  the  interpolation,  the  remaining  or  alternate  values 
of  8'  are  filled  in  by  direct  diff'erencing.  The  second  differences  are 
then  written  in  the  column  8",  their  regularity  proving  the  accuracy 
of  the  work. 

The  given  functions,  also  the  computed  first  differences,  etc.,  are 
distinguished  in  the  above  schedule  by  heavy  type. 

When  it  is  necessary  to  take  account  of  4th  and  5th  differences, 
we  have  only  to  form  an  extra  column  y,  to  follow  jS  in  the  schedule 
above.     Under  y  we  write  the  terms 


3    A^„+'^\  3    M+f/, 


128  V      2     y    '    128 


,  etc.; 


the  vahies    of  8'    are    then  formed    by  adding  the    three  corresponding 
terms  in  a,  /3,  and  y. 

Example.  — Given  the  values  of  log  sin  T  for  r=  30°,  32°,  34°, 
....  42°  ;  find  the  value  for  every  degree  of  T  from  32°  to  40°, 
inclusive. 

In  accordance  with  the  method  above  outlined,  we  arrange  the 
given  functions,  with  their  differences,  as  follows: 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


83 


T 

I^og  sin  T 

8' 

8" 

a 

(3 

J' 

J" 

J"' 

0 

30 

9.69807 

31 

+  2524 

32 

33 

9.72421 

9.73611 

+  1190 

1143 
1103 
1063 
1024 

988 
953 

+    920 

-45 

+  1107.5 

+  22.4 

2335 

-189 

+  20 

34 

9.74750 

42 

169 

35 

9.75859 

40 

1083.0 

1{\.1 

2166 

15 

3(i 

9.70922 

39 

154 

37 

9.77946 

36 

1000.0 

ls.:$ 

2012 

15 

38 

9.7S934 

35 

139 

39 

9.79887 

-33 

+  930.5 

+  10.7 

1873 

+  10 

40 

9.S0S07 

-129 

41 

+  1744 

42 

9.82551 

Since  -Itli  differences  may  be  neglected,  only  the  two  columns  a  and 
/8  are  required  for  the  computation  of  the  differences  S'.  All  the  quanti- 
ties actually  used  in  the  process  are  given  in  the  above  table.  The 
computed  quantities,  together  with  the  giveii  values  of  log  sin  T,  are 
printed  in  heavy  type,  to  render  this  process  more  evident. 

51.  To  Reduce  the  Argument  Interval  of  a  Given  Table  from  w 
to  1)10),  where  „  is  a  Positive  Odd  Integer. — As  particular  cases  of  this 
problem,  we  may  take  m  =  .] ,  i  ,  i,  etc.  Taking  ni  =  |,  we  intro- 
duce two  values  between  every  two  adjacent  functions  of  the  given 
table;    we  thus  derive  the  series 

F    F    F.    F    F 

in  which  the  interval  is  |w.  This  process  is  called  interpolation  to 
thirds.  To  interpolate  to  fifths,  we  let  ni  =  l ,  thus  introducing  four 
functions  between  every  two  adjacent  terms  of  the  original  series. 
"VVe  then  have  the  tabular  values  of 

F     F    F.    F,    F    F    F, 

o'S's'J'J'i'i'     ■     ■     ■     • 


the  interval  being  |w. 

More  generally,  let  us  take  m  =  ,,  where  k  is  a  positive  odd 
integer;  we  thus  introduce  /■  —  1  equidistant  values  of  the  function 
between  every  two  adjacent  terms  of  the  given  series.  The  resulting 
series  will  therefore  be 


F     F      F       F 


F  F 

^  (A— Dm)    -^  1; 


F,^ 


84  THE  THEORY  AXD  PRACTICE  OF  INTERPOLATION. 

in  whic'li  the  argument  interval  is  mm,  or  7^.  IN^ow,  the  two  adjacent 
functions  of  this  interpolated  series,  which,  as  a  pair,  fall  itiidica//  be- 
tween Fo  »nd  F^ ,  ai-e 

F  and      F 

that  is 


Hence,  if  we  put 


"m  """  "('?) 


S,'  =   F  —  F  C13n 


it  follows  that  S,'  is  the  value  of  the  Jirst  difference  of  the  interpo- 
lated series  which  falls  on  the  line  midwatj  between  F^  and  Fj ;  we 
shall  designate    this    quantity  a  ntiddle  first  diff'eretire   of  the    required 

series.     If  we  now  let 

1  +  m 

—2-  =  '*  (132) 

we  have 

1  —  m 

-2-   =  ^-'' 
and  (131)  becomes 

8,'  =   F,,-F,_,,  (133) 

Hence,  to  express  8,'  in  terms  of  the  differences  of  the  given  series, 
we  have  only  to  express  the  values  of  i'^,,  and  i^i_„  by  Bessel's 
Formula;    thus,  abbreviating  coefficients,  we  have,  as  in  (113), 

F„  =   7';  +  ««i  +  B/j  +  Cc,  +  I  hi  +  Ee,+   .    .    .    .  (134) 

Also,  by  virtue  of  the  property  of  these  coeificients  established  in  §41, 
we  have 

F,_„   =    F„+  (l-)i)a,  +  Bb  -  Cc,  +  D,I  -  Ee,+   ....  (135) 

The  difference  of  these  equations  gives 

8,'  =    F„-F,_„  =   (271 -1)  a, -\- 2  Cc,  + 2 Ee,+   ....  (136) 

Now,  by   (132),  we  have 

1  +  m 
n   =   — TT — 


hence,  from  (111),  we  find 

C  =   ln{n-\){n-~\)     =    ^  ('«—!) 

E  =   ^\^{n  +  \)n{n-\){n-2){n-\)   =    ^^{n\■\)  (n-2)C  =  _^  (,„^-l)(,«=_9) 


THE    THEORY    AND    PRACTICE    OF    INTKIU'OLATION. 


85 


Substituting-   these    values    of  ii,    C,   and  £J   in    (13G),    we    obtain    tlu- 
f'oruuila 


85'   =    wa,  +  !^  (/'r-l)  '\  +  j^^  (jn'-l)(„r-d)  e,  + 


(1-) 


by    Avliich    the   iiilddlc    first   differnuTs   may  be   computed    in   any  ease, 
|)ix)vided     —     is  a  i)ositive  odd  integer. 

Let  ns  now  consider  the  schednle  below  : 


T 

Fcn 

8' 

8" 

8'" 

J' 

J" 

Jill 

Jiv 

Jv 

t—iO 

F-, 

8'-' 

8^. 

8::; 

a' 

b' 

(-■' 

d' 

e' 

t  —  III  10 

t 

f  +  mm 

i^o' 

s/ 

80'' 

sr' 

"1 

K 

'■] 

'/o 

''1 

<  +  0)  —  IIIW 

t  +  co 

F,' 

•  • 

8/' 

b: 

'': 

The  quantities  are  here  arrangx'd  in  a  manner  somewhat  similai- 
to  the  schedule  of  §50.  The  given  functions,  F_,,  F„,  Fj,  .  .  .  .  , 
are  separated,  snccessively,  by  h —  1  blank  lines  or  spaces,  for  the 
subsequent  entry  of  the  interpolated  values.  The  columns  8',  8",  and  8"' 
are  also  reserved  for  the  dilierences  of  the  interpolated  series  ;  and 
the  diiferences  of  the  given  functions  are  written  to  the  right,  in 
columns  J'  to  J'\ 

The  value  of  8,'  is  now  computed    by  (137)   from   the    ditfercnces 
rti,  ('i,  and  ("1,  which  stand  opposite.      In    like   manner,  8'^;  is  computed 
from  the  ditferences  a,  c,  and  e  ;    8/,  from  a,,  Cg,  and  e.^;    and   so   on.  ^ 
We  thus  obtain  a  series  of  middle  first  di.ferences,  which  are  tal)ulated 
under  8'  in  the  schedule  above. 

Now  it  is  clear  that  if  we  should  interpolate  the  A-  —  1  inter- 
mediate terms  between  8'_,  and  8,',  between  8,'  and  8/,  etc.,  the 
resultinsf  series  would  constitute  the  consecutive  first  diflferences  of 
the  inter2wlated  series  F (T)  ;  the  required  functions  would  then  be 
formed   by  successive  additions   of  these  difierences.     The  problem  of 


S() 


THE    TIIEOllY    AND    PKACTTCE    OF    INTERI'OLATION. 


interpolating-  tlie  given  sei'ies  F {T)   is  thus  vii'tually  lediiced    to   tliat 
of  interpolating  the  comjnited  values  of  S'  in  precisely  the  same  )ii<(inicr. 

Now,  let  S„"  denote  the  second  differenee  of  the  Interpolated  series 
F,  which  stands  opposite  Fq]  8,",  the  second  difference  opposite  F^; 
etc.  It  follows  that  80"  is  the  middle  first  difference  of  the  interpolated 
sei-ies  8',  which  falls  between  8'_^  and  8',  ;  8/',  that  falling  between 
8/  and  8';  and  so  on.  Hence,  we  may  find  8^",  8/',  82",  ....  from 
the  computed  series  8'_, ,  8,',  8,',  .  .  .  .  ,  in  precisely  the  manner 
that  the  latter  quantities  are  derived  from  F_i,  Fi,,  F^,  .  .  .  .;  that 
is,  l)y  application  of  the  general  formula  (137),  mutatis  mutandis.  For 
this  purpose,  we  must  form  the  ditferences  of  the  computed  series 

8'_,,  W,  8,',    .... 
Accordingly,  let  us  ]nit,  for  brevity. 


and   (137)  becomes 


-1) 


^^'=    UVH.^'""-^)i"'"-^) 


8,'  =   ,iia,+  M'\+  31' Ci 


(1.3S) 


(139) 


provided  differences  beyond  J'   ai-e  disregarded.     We  now  form  a  table 
of  the  quantities     8'_,,  8,',  8/,  .  .  .  .  ,     and    their  differences,  as  follows  : 


Function,  =  8' 

1st  Diff. 

2d  Diff. 

3d 

4th 

8'.,  =  ma'  +  3[r'+  31' e' 
S/  =  ma,  +  3Ic,+  3I'ei 
8','  =  ma,  +  3Ic^+  31' e.. 

mb'  +  3fd' 
mh^  +  31  d^ 
mb,  +  3Id, 

mc'  -1-  Me' 
mc^  +  3Ie, 

vuV 
mdo 
mdi 

me' 
me, 
VI  e„ 

Whence,  applying  the  general  formula  (130)  to  the  quantities  of   this 
table,  we  obtain 

8/'   =    m  (m\  + 3Id,;)  +  31  (md^)   =   m-%  +  23Imd^ 

or,  by  (138), 

So"   =    m\+'^{m."--l)d,  (140) 

by  which  the  quantities     8"_, ,  8„",  8,",  ....       of   the  former   schedule 
are  computed  from  the  differences  J"  and  .1"  which  stand  opposite. 

Again,  we  may  suppose  that  the  intermediate  \alues  of  8"  have 
been  interi)olated  between  the  computed  values  8"_,,  So",  8,",  .  .  .  .  ; 
this  completed  series  8"  constitutes    the   consecutive  second  differences 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATIOX. 


87 


()(■  the  interpolated  scries  F(T).  Finally,  wc  sliall  denote  by  8'"  the 
third  diiference  of  the  interpolated  series  F,  which  stands  ojjposite 
S,'  in  the  given  schedule.  The  (juantity  8'"  is  therefore  the  middle 
iirst  difference  of  the  completed  series  8",  whicli  falls  between  8„"  and 
8,";  it  bears  the  same  relation  to  8o"  and  8/',  that  8,'  bears  to  7'^,  and 
Fi .     Hence,  to  find  SJ" ,  let  us  i)ut 


31"   =   ^(..^-1, 


and   (140)   becomes 


So" 


/«%„  +  iI/"(/„  (141) 

The  diiferences  of    8"_i ,  8u " ,  8i " ,  ....     are  therefore   as  follows: 


Function,  =  8" 

1st  Diff. 

2(1 

3d 

8",  =  m%'  +  M"J' 

So"  =  v^\  +  ^I"'h 
8,"  =  m%  -1-  MVcl, 

mh:'  +  M"e' 

niH' 

Whence,  applying  (as  above)  the    general  formula   (130),  we  find 

8;"   =   m{m\  +  M"ei)-¥M{w,\)    =   m^\+ {wU"  ^- m-M)e^ 

8ubstituting  the  values  of  M  and  M" ,  we  have 

81"  =   '"%+"''("'^-l)e,  (142) 

O 

In  practice,  the  values  of  8'"  and  8'  are  never  required,  and  in 
many  cases  the  column  8"'  is  not  necessaiy.  Supposing,  however,  that 
we  have  computed  the  (nearly  constant)  values  of  S"!  ,  8,  ",  81",  .  .  . 
by  (142),  the  intermediate  terms  ai"c  then  written  in  by  mere  inspec- 
tion. We  thus  complete  the  column  8'", — the  consecutive  third  diffei*- 
ences  of  the  required  series  F{T).  Having  also  computed  the 
quantities  S,,",  8,",  8^",  ....  and  8'_j,  8,' ,  8/ ,  .  .  .  .  ,  we  complete 
the  columns  8"  and  8',  and  hence,  also,  the  interpolated  series  F(T), 
by  successive  additions. 

We  now  bring  together  the  formulae  for  8/,  S^",  and  8^",  in  the 
order  computed  in  practice,  as  follows  : 


S;"   =   m^;+-(m'-l)e, 

So"  =  »^\  +  '^(^»''-^yio 

8/     =   ma,  +  ^  (m"--  1)  c,  +  j^^  (m"--l)(vi--9)  e, 


(143) 


S8 


THE    THEORY    AND    TKACTICE    OF    INTEl!lH)LATION. 


which  serve  to  reduce  the  tabulai'  interval  to  iii  times  its  original 
value,  ni  being  the  reciprocal  of  a  positive  odd  integer.  It  will  he 
observed  that  the  differences  re(|uiri'd  in  computing  each  of  the  (piantities 
8  are  always  found  on  the  same  line  with  that  (jnantity. 

52.     fiit('rpolatio)i,   to    Thirds.  —  For  tliis  i)ur])ose,  we    take    vi  =:  ;| 
in   tlie  foi'nuilac   (1^:3),  and   find 

v  =  j''o-?i3<  y  (144) 

These  formulae  are  more  conveniently  computed  in  tlie  form 


s;"  =  ,',('• 


«i) 


(145) 


S/    =  M«i-8i") 

Example.  —  Given  the  value  of  log  tan  T  for  every  third  degree 
of  T  from  27°  to  48°,  inclusive  :  find  the  function  for  every  degree 
between  33°  and  42°. 

According  to  the  precepts  of  the  last  section,  we  arrange  the 
work  as  follows  : 


r 

Log  tan  T 

8' 

8" 

S'" 

J' 

J" 

J'" 

Jiv 

o 

27 

9.70717 

+5427 

30 

9.7«144 

+  3.1 

5108 

-319 

+  85 

33 

34 
35 
36 

9.S12r)2 

9.82899 
9.84523 
9.86120 

+  1646.9 
1623.8 

1603.3 
1585.3 
1569.6 

1556.1 
1544  6 

-25.9 

23.1 
20.5 
18.0 

O.O 

2.8 
2.6 
2.5 
2.3 
2.2 
2.0 
0  0 

4874 

234 
163 

71 

-14 
12 

37 
38 
39 

9.87711 
9.89281 
9.90837 

15.7 
13.5 
11.5 

4711 

104 

59 

6 

40  ■ 

41 

42 

9.92382 
9.93917 
9.9)444 

1535.1 

+  1527.5 

9.5 

7.6 

-   5.7 

1.9 

1.9 

1.9 

+  1.9 

4607 
4556 

-   51 

53 

+  51 

_   2 

45 

0.00000 

+  4556 

0 

48 

0.04556 

The  heavy  type  shows  at  a  glance  the   given  functions,  and  like- 
wise  the   computed  middle   differences.      We   observe   that   it    is    here 


THE    THEORY    AND   I'KACTICE    OF   INTEKPOLATION.  81) 

necessary    to    compute  five    values  of   8'",  foiii'  values    of  S",  and    only 

three   of  8'.      These    quantities    are    computed    to    one    more    than    the 

number    of  decimals    given    in    F(T),    to    avoid    accumulation    of  any 

appreciable    ci'ror    in    the   linal    additions,     liaving   obtained   for  8"  tlie 

series 

+  3.1         2.6         2.2         1.9         +1.9 

the  intermediate  terms  are  readily  inserted,  as  shown  aliove;  it  is 
necessary,  howevei',  to  see  that  the  completed  series  8"  is  consistent 
with  the  compated  values  of  8".     Thus  we  must  have 

2.8  +  2.6  +  2.5   =    -(18.0-25.9)   =    +7.9 

2.3+2.2  +  2.0   =    -(11.5-18.0)   =    +6.5 

•  2.0  +  1.9  +  1.9   =    -(  5.7-11.5)   =    +5.8 

If  these  relations  ai'e  not  satisfied  exactly  on  first  trial,  the  interpo- 
lated values  of  8"'  must  be  adjusted  to  fulfill  the  necessary  conditions. 
The  column  8"  is  now  completed  by  successive  additions  of  the 
quantities  8".  Again,  it  is  necessary  to  see  that  the  completed  sei'ies 
8 "  agrees  with  the  computed  values  of  8'.  For  we  must  have 
-(20.5  +  18.0  +  15.7)   =   15G9.6  -  1623.8   =    -54.2,  etc. 

Since  these  relations  are  seldom  exact  in  the  beginning,  the  pro- 
visional values  of  8"  will  usually  require  slight  alterations. 

From  the  final  series  8",  we  obtain  8'  by  successive  additions.  As 
before,  an  agreement  must  subsist  between  the  values  of  8'  and  the 
given    set    of  functions  ;    that  is,  between  8'  and  j'.     Thus  we   should 

have 

2"8'  =   1646.9  +  1623.8  +  1603.O   =    +4874.0   =   J',  etc. 

In  the  latter  case,  however,  a  discrepancy  not  exceeding  four  or  five 
units  in  the  added  decimal  may  be  tolerated.  Our  final  series  8'  is 
therefore  satisfactory  ;  whence  we  obtain  by  successive  additions  the 
required  values  of    log  tan  T. 

53.  Interjwlatioti  to  Fifths.  —  Taking  ■))i  =  I  in  the  formulae 
(143),  we  obtain 

V  =    AC^o-A'^o)  }  (146) 

In  practice  it  will  suffice  to  put  ^e^  for  both  rf^e^  and  yVs^ij  ^^^ 
formulae   (liO)  then  become,  vei-y  approximately, 


Ur 


DR.  GEORGE  K  iVicEWE^=. 


00 


THE    TIIEOKY    AKD    PKACTICE    OF   IXTKRI'OLATION. 


8,'  =  itti-s;" 


(147) 


Example.  —  The  following  ephemeris  gives  the  moon's  K.A.  for 
every  ten  hours.  Obtain  the  value  for  every  second  hour,  from 
Sept.  23''  20"  to  Sept.  25''  12",  inclusive. 

The  details  of  the  computation  are  as  follows  : 


Date,  1898 

Moon's  R.A. 

8' 

8" 

8'" 

J' 

J" 

J'" 

Jiv 

il        h 

h        111       8 

m       s 

s 

s 

ni     s 

B 

s 

8 

Sept.  23     0 

IS  24  20.4 

+  25  31.1 

Sept.  23   10 

IS  4!>  57.5 

-.034 

32 

25  11.0 

-20.1 

-4.2 

Sept.  23  20 

1»  15     S.5 

+  4  59.39 
58.39 

4  57.36 
5  0.31 
55.23 
54.13 
5  3.02 

4  51. SO 
50.75 
49.60 
48.44 
47.28 

4  46.12 
44.95 
43.79 
42  63 

-0.076 

30 

28 

26 

.024 

23 

20 

1  7 

24.3 

+  1.2 

23  22 

24  0 
24     2 

19    20      7.9 
19   25      6.3 
19    30      3.7 

1.004 
1.0  30 
1.054 

24  46.7 

3.0 

24     4 
Sept.  24     6 

19    35      0.0 
10  30  55.2 

1.077 
1.097 

27.3 

1.5 

24     8 
24   10 
24    12 

19    44    49.3 
19   49    42.3 
19    54    34.2 

1.114 
1.128 
1.140 

J- 1 

14 

.012 

09 
07 
05 
03 
-.001 
+  .002 
03 
06 

24  19.4 

1.5 

24    14 

19   59    25.0 

1.149 

Sept.  24  1« 
24    18 

20     4  14.0 

20      9      3.0 

1.156 

1.161 

28.8 

1.4 

24    20 

24  22 

25  0 

20    13    50.3 
20    18    36.4 
20   23    21.4 

1.164 
1.165 
1.163 

23  50.6 

-0.1 

Sept.  25     2 

20  2S     5.2 

1.160 

28.9 

1.1 

25      4 

20   32    47.8 

41.47 
4  40.33 

39.19 
1-4  38.06 

1.154 

07 
.OOS 

09 
11 
12 

25     6 
25     8 

20    37    29.3 
20   42      9.G 

1.147 
1.139 

23  21.7 

+  1.0 

25    10 
Sept.  25  12 

20    4G    4S.S 
20  51  2().0 

1.130 
-1.110 

27.9 

+  0.9 

14 

+  .015 

22  53.8 

+  1.9 

Sept.  25  22 

21    11  20.7 

+  22  27.8 

-26.0 

Sept.  26     8 

21  36  48.5 

THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


91 


Here  we  extend  the  computation  of  8  "  and  S"  two  j>laces  of"  deci- 
mals ;  one  of  which  is  dropped  in  computing  8',  and  the  otlier  in 
forming  the  required  functions.  The  principle  and  method  being  the 
same  as  in  the  last  example,  further  explanation  is  unnecessary. 

54.  Order  of  Iiiterpolatioit,  to  Follow,  wJien  a  Series  lieqaires 
Successive  Interpolation  to  Halves,  Thirds,  etc  —  When  a  table  of 
functions  is  to  be  interpolated,  successively,  one  or  more  times  to 
halves,  and  also  to  thirds  and  Jifths,  the  easiest  method  is  to  proceed 
in  the  order  named.  Thus,  if  the  interval  of  the  original  series  is  w, 
and  that  of  the  final  table  is  w,  we  may  suppose  the  relation  of  these 
quantities  to  be  — 

u,  =  2'.3'.5'".<«' 

where  1-,  I,  and  )n  arc  integers.  It  will  then  be  found  most  expedient, 
first,  to  interpolate  to  halves,  k  times  ;  then  to  thirds,  I  times  ;  and 
finally  to  fifths,  m  times. 

For  example,  F  being  given  for  every  degree,  and  requii-ed  for 
every  minute  of  arc,  we  should  first  interpolate  to  30',  then  to  15', 
then  to  5',  and  finally  to  every  minute  of  arc. 

55.  To  Interpolate  with  a  Constant  Interval  n,  an  Entire  Series 
of  Functions.  —  Let  the  given  series,  with  its  differences,  be  as 
follows : 


T 

F(T) 

J' 

J" 

J'" 

Jiv 

f. 

t    +    (U 

t  +  2^ 
f +  3<o 
t  +  4(0 

F, 
F, 
F, 

a„ 

«4 

0 

"1 

"3 
"4 

'■1 

'■3 
''4 

It  is  required  to  interpolate  the  values  of  F„,  Fi_^_„,  I\^n,  i^3+„,  .... 
These  functions  evidently  form  a  new  series  having  the  same  interval 
as  the  old.  Let  us  denote  this  new  scries  by  [i'^J;  also,  let  the  dif- 
ferences of  [i^],  denoted  by  [j],  [j"],  [J"'],  .  .  .  .  ,  be  taken  as 
shown  in  the  table  below  : 


92 


THE    THKOHY    ^VXl)    rUACTICE    OF    INTEIU'OLATION. 


T 

[F] 

[J'] 

[J"] 

[-!'"] 

[Jiv] 

t  +  nu> 

t+  (l  +  ii)i 

t  +  (2  +  h)w 

f  +  (3  +  7l)a) 

t  +  (4  +  ?i)<" 

F,^n 

F 

«1 
lt.„ 

«4 

A, 

y^ 
y!i 
y4 

Now,  it  was  shown  in  §22  that  differences  of  (inii  order  may  be 
expressed  in  terms  of  the  tabular  functions.  Thus,  in  pai'ticuhu-,  we 
obtain  from  the  given  series  F, 


r.,  =  F,-  3F„_  +  3F,  -  i^„    =   *  (t+u,) 
r,   =   F,  -  3F.,  +  3F„  -  f[     =   *(/'  +  2u,) 


(148) 


where     "^1^  (/)     denotes,    for    Ijrevity,    the    function    of    /    expressed    by 

F„  -  3F^  +  3/;  -  F_, ;     that  is,     *  (t)  =  F(t  +  2,o)  -  3F(t  +  o>)  +  3F(l)  -  F(t-w) 
Again,  in  Hke  manner,  the  interpolated  series   [i'^]   gives 


=   F,, 


3/'':+,,  +  3i^„ 


F 


y„  =   F.^.„  -  3F„^„  +  3F^^„  -  F„      =   *  (i  +  «,  +  Ma,) 


(140) 


It  follows,  then,  that  the  series  [/J'"]  is  simply  the  scries  z/'"  inter- 
polated forwaid  with  the  constant  interval  n.  Moreover,  since  the 
above  reasoning  is  perfectly  general,  this  relation  holds  for  an;/  order 
of  differences. 

Hence,  to  perform  the  required  interpolation  of  the  series  F(T), 
that  is,  to  obtain  the  series  [i^],  we  have  only  to  interpolate  forward 
each  value  of  J"  with  the  constant  interval  >/,  thus  forming  the  column 
[j"].  This  process  is  obviously  brief  and  simple.  Then,  if  we  com- 
pute occasional  values  of  [j'],  and  also  of  [-F'],  we  readily  complete 
the  required  table  by  successive  additions,  as  in  the  preceding  problems. 

Example.  —  To  illustrate  the  process,  we  tabulate  the  "Latitude 
Reduction"  for  evei-y  fourth  degree  of  latitude  (g)  from  30°  to  82°, 
and  thence  derive  the  series  for  <r  =  35°,  39°,  43°,  ....  75°.  The 
work  is  arranged  as  follows  : 


THE    TIIEOKY    AND    PRACTICE   OF   INTERPOLATION. 


93 


<F 


30 
34 
38 
42 
46 
50 
54 
68 
G2 
66 
70 
74 
78 
82 


-f 


605.56 
648.60 
679.06 
696.34 
700.08 
690.19 
666.84 
630.47 
581.78 
521.70 
451.40 
372.24 
285.77 
193.69 


+  43.04 
30.46 
17.28 

+  3.74 

-  9.89 
23.35 
36.37 
48.69 
60.08 
70.30 
79.16 
86.47 

-92.08 


-12.58 
13.18 
13.54 
13.63 
13.46 
13.02 
12.32 
11.39 
10.22 
8.86 
7.31 

-  5.61 


J'" 


-0.60 
0.36 

-0.09 

+  0.17 
0.44 
0.70 
0.93 
1.17 
1.36 
1.55 

+  1.70 


JIv 


+  .24 
.27 
.26 
.27 
.26 
.23 
.24 
.19 
.19 

+  .15 


[<f-V'] 


3.J 

39 
43 
47 
51 
55 
59 

67 
71 
75 


684 
698 
698 
685 
fiaS 
619 
567 
505 
432 
351 


.422 

.634 
.552 
,883 
,602 
.}>45 
420 
785 
032 
,382 
,244 


[-)'] 


+  27.212 

13.918 

+    0.331 

-13.281 

26.657 

;j«.525 

51.635 

62.733 

72.650 

-Sl.KJS 


[J"] 


13.294 
13.5S7 
13.«12 
13.37« 
I2.S«S 
12.110 
11. lis 
».S«7 
S.4S8 


Taking  n  =  0.25,  we  compute  by  Bessel's  Formuhi  tlK'  value.s 
of  v  —  l'  for  <7  =  35°,  55°,  and  75°,  extending  the  decimal  one  unit. 
Similarly,  we  compute  fJiree  values  of  [j'],  and  all  of  [J"];  the  com- 
puted quantities  being  clearly  shown  by  heavier  type.  Adjusting 
slightly  the  series  [j"]  to  conform  to  the  computed  values  of  |  yj,  we 
complete  the  latter  column  by  successive  additions.  The  values  of 
[j']  being  found  to  accord  with  the  computed  functions,  we  complete 
the  entire  series  as  required. 

Since  the  computed  intermediate  values  of  [.J']  and  [i^]  serve  only 
as  checks,  it  is  obvious  that  their  positions,  as  also  the  intervals  of 
their  distribution,  are  entirely  arbitrary.  These  are  details  to  be  decided 
by  the  computer's  judgement  in  any  given  case. 

It  may  occasionally  be  practicable  to  extend  the  ])rocess  to  the 
computation  of  [J'"]. 


94 


THE    THEOnY    AND   PRACTICE    OF    INTERPOLATION. 


EXAMPLES. 

1 .  Tabulate  the  five-place  log  cosines  of  15°,  18°,  21°,  24°,  27°,  80° ; 
fi-oin  these  values  interpolate  log  cos  T  for  T  =:  17°  43',  23°  8',  and 
28°  15',   respectively. 

2.  Given  the  following  tal)le  : 


T 

F(T) 

T 

F(T) 

10 
20 
30 

17"31 
14.68 
13.62 

40 
.50 
60 

14.16 
16.34 

20.18 

Compute   tlie   values    of   F  for    T  =  24.(>,  28.8,  32.3,  and  48.5,   using 
either  Bessel's  or  Stirling's  Formula. 

3.  Interpolate  the  required  functions  of  Example  2  by  means  of 
a  corrected  first  <lifference,  as  explained  in  §§44  and  45. 

4.  What  is  the  maximum  ei-ror  of  interpolation  in  the  table  of 
Example  2,  supposing  that  second  differences  are  neglected  ? 

5.  Find  the  correct  values  of  the  erroneous  functions  in  the 
several  tables  of  Example  (3,  Chap.  I,  by  direct  interpolation,  as  ex- 
plained in  §§4(3  and  47. 

(5.  Given  the  following  twelve-hour  ephemeris  of  lunar  distances 
of  Splca  : 


Date 

L.D.  of 

Date 

L.D.  of 

1898 

Spica 

1898 

Spica 

<\ 

O           /           It 

d 

o        r       n 

•July  1.0 

43  24  9 

July  3.0 

73  35  46 

1.5 

50  52  0 

3.5 

81  12  52 

2.0 

58  24  0 

4.0 

88  48  56 

July  2.5 

65  59  1 

July  4.5 

96  22  40 

THE    THEORY    AND    PUACTICE    OF    INTERPOLATION", 


95 


Interpolate  the  series  tivicc  to  halves;  the  first  result  to  include  tin; 
values  from  July  1''.5  to  4''.(),  and  the  final  three-hour  ephemeris  to 
extend  from  July  2''  0''  to  July  :t'  12",  inclusive. 

7.     The  ephemeris  below  gives  the  sun's    true  longitude  for  eveiy 
third  day  : 


1898 


Oct.    7 

10 

Oct.  13 


Sun's  Longitude 

1898 

194°14'35'!2 
197  12  34.2 
200  10  54.0 

Oct.  16 

19 

Oct.  22 

Sun's  Longitude 


203  9  32.9 
206  8  29.4 
209  7  42.0 


Derive  from  these  values  a  dailjj  ephemeris  extending  from  Oct.  10  to 
Oct.  19,  inclusive. 

8.     The    following    table    contains    the    heliocentric    longitude    of 
Japiter  for  every  80th  day  of  1898-99,  beginning  with  Jan.  0,  1898  : 


Date 

Helioc.  Long. 

Date 

Helioc.  Long. 

1898 

of  Jupiter 

1898 

of  Jupiter 

i\ 

0           f              II 

d 

O           1               II 

0 

178  59  17.9 

320 

203  10  20.5 

SO 

185     2  24.1 

400 

209  13  53.8 

IGO 

191     5     0.3 

480 

215  18  35.1 

240 

197     7  30.9 

560 

221  24  48.1 

Interpolate  this  table  to  halves,  extending  the  series  from  120''  to  llO'' 
inclusive;  designate  this  forty-day  ephemei'is,  Table  A.  Then  inter- 
polate A  to  fifths,  denoting  the  eight-day  series  by  B.  Let  the  limits 
of  B  be  200''  and  320'',  respectively.     Retain  copies  of  A  and  B. 

9.  Interpolate  (forward)  the  longitudes  of  Table  A,  Example  8, 
with  the  constant  interval  ii.  ^  0.20,  by  the  method  of  §55.  This 
will  furnish  an  ephemeris  for  the  dates  1(58'',  208^  248'',  ....  368''. 
Compare  the  longitudes  thus  found  foi'  208'\  218'',  and  288'',  with  their 
values  in  Table  B,  Exanii^le  8. 

10.  Deduce  from  the  general  formulae  (113),  the  special  formu- 
lae for  interpolation  to  sevenths.     Make  an  application  to  the  five-figure 


96  THE  THEORY  ANT)  PRACTICE  OF  INTERPOLATION. 

logarithms  of    47,  o-t,  01,  ....  9(5,    by  computing-    the    logarithms    of 
the  consecutive  numbers  between  (il    and  ^T). 

11.  Show  that  if  the  formulae  (143)  wei-e  extended  to  include 
the  middle  differences  of  order  /,  we  should  iiave  (using  the  symbolic 
form  of  notation  employed  in  the  analogous  foiMuulae  (04)) 


in  which  i  may  be  either  odd  or  even  ;    and  where     J',  J'-^-,  z/'+\    .... 
symbolize  the  tabular  differences  wliich   fall    upon    the   same  horizontal 
line  Avitli  S'. 


CnAPTER  III. 

DERIVATIVES   OF   TABULAR   FUNCTIONS, 

56.  It  is  often  required  to  find  certain  numerical  values  of  the 
differential  coefficients  of  functions  either  analytically  unknown,  or 
complicated  in  exj^ression.  In  the  majority  of  such  cases  the  function 
has  been ,  previously  tabulated  for  i^ailicular  (equidistant)  values  of 
the  argument.  The  required  derivatives  are  then  readily  computed 
from  the  differences  of  the  tabular  functions. 

We  have  already  seen  that  —  with  certain  limitations  —  particular 
values  of  a  function,  with  their  differences,  practically  determine  the 
character  and  law  of  that  function,  thus  enabling  us  to  determine 
intermediate  values  by  interpolation.  The  trend  or  law  of  variation 
of  the  function  being  thus  defined  by  its  difierences,  it  is  but  natural 
to  suppose  that  the  successive  derivatives  are  quantities  closely  related 
to  these  differences  ;  since  the  derivatives  are  themselves  direct  in- 
dices of  the  character  of  variation  of  the  function. 

57.  Practical  Applications.  —  The  most  useful  application  is  in 
finding  the  change  or  variation  ini^(T)  corresponding  to  an  increase 
of  one  unit  in  T,  supposing  the  rate  of  change  in  F  to  remain  con- 
stant from  T  to  T-|-l,  and  equal  to  the  actual  rate  at  the  instant  T; 
for  this  quantity  is  simply  the  first  differential  coefficient  of  F'(T) 
with  respect  to   T,  which  we  shall  denote  by  F\T). 

For  example,  having  obsei'ved  that  a  freely  falling  body  describes 
sixteen  feet  during  the  first  second  of  its  descent,  forty-eight  feet  the 
second  second,  and  eighty  feet  the  third,  its  velocit;/  at  the  end  of 
two  seconds  is  easily  found  to  be  sixty-four  feet  per  second.  This 
velocity  of  sixty-foiu*  feet  is  nothing  more  than  the  first  difierential 
coefficient  of  the  space  with  respect  to  the  time,  comjjuted  for  the  in- 
stant 2^0  :    it  is  tlie  space  which  would  be  described  during  the  third 


98         THE  THEORY  AND  PRACTICE  OF  IXTERPOLATION. 

second,  supposing  the  action  of  gravity  to  have  ceased  at  the  end  of 
the  second  second. 

The  most  frequent  and  important  apphcations  occur  in  Astronomy. 
An  astronomical  ephemei'is  contains  a  great  variety  of  tables  giving  the 
positions  and  motions  of  various  heavenly  bodies,  and  of  certain  ])oints 
of  reference.  From  the  given  positions,  tabulated  for  every  hour  or  from 
day  to  day,  are  derived  the  motions  per  minute,  jier  hour,  or  per  day. 
according  to  circumstances.  For  instance,  the  JVatitical  Almanac  gives 
the  sun's  declination  for  every  Greenwich  noon.  The  hourlij  motion 
in  declination  (also  given  for  every  noon)  is  computed  from  the  dif- 
ferences  of  the  tabulai-  declinations  :  its  value  is  the  differential  coef- 
ficient of  the  tabular  function  at  tlie  date  in  question. 

In  the  following  sections  the  various  formulae  employed  in  com- 
puting the  derivatives  of  tabular  functions  will  be  derived. 

58.  Development  of  the  Required  Formidae  in  General  Terms. — 
The  variables   T  and  n  ai-e  connected  by  the  fundamental  relation 

T  =   t  +  nm  (150) 

in  which  t  and  w  are  constants  for  a  given  series.  Accordingly,  we 
have  hitherto  written,  under  varying  circumstances. 

F{T)     ,     F{t  +  nw)     ,     F„ 

as  equivalent  expressions  of  the  same  quantity.  In  like  manner,  Ave 
shall  hereafter  denote  the  successive  derivatives  of  F {T)  by  the  fol- 
lowing equivalent  forms : 


dT 

_d 

d 


^  \  F(T)  I    =  F'  (T)  =  Fi    {f  +  nm)  =  F'„ 
^j-,^F{T)  ^    =  F"(T)  =  F"  (<  +  «o>)  E  F': 

When  it   is    convenient   to   proceed  hachwards  from  the  argument 
/  with  the  interval  n,  we  .shall  use  the  expressions 

F'_„  =  F'(i-no>)     ,     F'l„  =  F"(t-n^)    ,      F'l'„  =  F"'(t-n.o)     , (152) 


THE  THEORY  AND  PKACTICE  OF  INTERPOLATION. 


99 


Now,  by  means  of  any  one  of  the  fundamental  formulae  of  inter- 
polation, we  may  expi-ess  F^  in  the  form 


F,^  =   /;  +  na  +  Bh  +  Cr  +  Dd  +  Ee  + 


(153) 


where,  in  any  given  case,  a,  h,  c,  .  .  .  .  are  known  differences; 
and  where  B,  C,  D,  .  .  .  .  are  definite  functions  of  71.  Let  the 
successive  derivatives  of  B,  C,  JJ,  .  .  .  .  ,  taken  with  respect  to  m, 
be  denoted  by 


B' 

,  B"  ,  B'"  ,    .    .    .    . 

C 

,  C"  ,  C"  ,    .    .    .    . 

£)' 

,  1>"  ,  B'"  ,    .    .    .    . 

E' 

,  E"  ,  E'"  ,     .    .    .    . 

Then,  observing  that  the   coefficient    of  J"'  is    always  of  the    degree  i 


in  n,  we 

have 

dB 

dn 

B' 

dC 
dn 

C" 

dl) 

dn 

= 

D' 

dE 

dn 

=   E' 

d^B 

dri' 

B" 

d'C 
dn^    ~ 

C" 

d-D 

dn-" 

= 

D" 

d^E 
dn- 

=  Ell 

d^B 
dii'    ~ 

0 

d'C 
dn^ 

C" 

d'D 
dn" 

= 

D'" 

d^E 
dn" 

=   Eiii 

d*C 
dn"    " 

0 

dW 

dn" 

= 

D" 

d"E 
da" 

=  E" 

d^D 

dn'' 

= 

0 

d'E 

dn'' 

d^E 

d7l^ 

=  E^ 

=  0 

(154) 


Reverting  to  (151),  we  have 


El    = 


IT 


dF„      dn 
~dn    '  dT 


(155) 


Prom  (150)   we  derive 


whence 


dn 
dT 


F'    = 


1      dE\, 
<u       dn 


(156) 


(157) 


DR.  GEO! 
100  THE    THEORY    AND   PRACTICE    OF   INTERPOLATION. 

In  like  manner  we  obtain 

dF'  dF' 

pit    _    "-^  "  "-^  • 

pin  _ 

n 

iTiv   =   'iH^  =  '11^      m   ^    i^      ((^  j  (158) 

FZ    = 


(ZT 

dn 

rf/^;; 

dF'J 

f/r 

dn 

dF',!' 

dF',!' 

dT 

dn 

dF^: 

dF': 

dT 

dn 

dn 

1 

d'F^ 

dT  ~ 

to" 

■    dii' 

dn 

] 

d^F„ 

dT  ~ 

oi»    ■ 

■     dn' 

dn 

1 

d'F„ 

dT  ^ 

z*  ' 

■    dn' 

dn 

1 

d'F„ 

dT 

a,^    ' 

■    (/?i» 

Therefore,  nsing  (153)   and   (151),  we  find 

F'„    =   -  («+7iY> +CV +  //(/+ i;^'e+  .    .    .    .) 

f;;   =  1  (i?"/y+c''v +//'<;+ ^■'"(!+ .  .  .  .) 

w 

F'"    =   ~„  (C"'c  +  I)"'d  +  F"'e+  .    .    .    .  ) 

/'i^    =    -.  (I)"d  +  E'^e+  .    .    .    .  ) 

i^v        =       1     (^v,+     .      .      .      .    ) 


(159) 


which  are  the  general  formulae  for  computing  the  derivatives  of  F{T) 
in  terms  of  the  tabular  differences. 

To  derive  the  formulae  for  F'^n,  F'!_„,  F'"„,  .  .  .  .  ,  that  is,  to 
find  the  successive  derivatives  of  i^(^ — «w),  we  have  only  to  alter 
slightly  certain  details  of  the  preceding  development,  as  follows  : 

(1)  For  equation  (153)  must  be  substituted  the  corresponding 
expression  for  i^_„,  which  has  the  form* 

.  F_„  =   F^  -  na  +  Bfi  -  Cy  +  D8  -  Ft+  ....  (160) 

where     a,  ^,  y are,  in    general,  different   fi'om    the    dift'erences 

a,  h,  c,  .  .  .  .     of  (153). 

(2)  In  the  present  case,  we  have 

T  =   t  —no> 

and  therefore 

dn  1 

'elf  ^    ~  w 

which  must  be  substituted  for  equation   (15G)  above. 

•Compare  (75),  (105)  and  (111")  with  (7.3),  (104)  and  (111),  respectively. 


THE    THEORY    AND   PRACTICE    OF    INTERPOLATION.  101 

Introducing  these  changes,  and  operating  as  before,  we  obtain  (he 
required  forinnhie,  namely, 

FL„  =  -  (a-n'/s+Cy-rn+E'^- .  .  .  .) 

F1„  =  I^(n"l3-C"y+/)"S-E"t+  .    .    .    .  ) 


F'l',^  =  \{C'"y-Dn  +  E"U-  .    .    .    .) 

i^'",.  =  4  (rPh-E"'f.+  .  .  .  . ) 

Fl,,  =   -,{E^c-  .    .    .    .  ) 


(lOI) 


It  now  remains  to  apply  (159)  and  (161)  specifically  to  each  of 
the  several  formulae  of  interjjolatiou,  of  which  (153)  is  the  general 
type.  It  is  obvious  that  a  particular  set  of  coefficients,  B',  B",  .  .  .  . , 
C,  C",  .  .  .  .  ,     etc.,  will  result  in  each  case. 

59.  To  Comjmte  Derivatives  of  F{T)  at  or  near  the  Beginning 
of  a  Series.  —  The  formulae  adapted  to  this  pin-pose  are  derived  from 
Newton's  Formula  of  interpolation  (73),  which  is  — 


where 


Fn  =   ^0  +  »'\  +  I'h  +  C^o  +  ^'''o  +  Er^^  .    .    .    .  ^  (1G2) 


__   n{ti  —  \)         n-       n 
^         ^   2  ~2 
_  _   n(n~l)(n  —  2)        n^       n^       n 
^  ii  ^  F~   2^  ■•■  3 

^  w(w-l)(n-2)(»-3)  ^  !^  _  ™'       11    o  _  w  V  ^jg3^ 

li  24       4   "'"24"       4 

m(w-1) (w-4)  w=        «^        7     3       „  ,. 

^  =  ^ =120-12  +  ^''   -19'^  +  n 


Differentiating  these  expressions  successively  with  respect  to  n,  as 
indicated  in  (154),  and  substituting  the  resulting  values  of  B\  B",  .  .  .  . , 
C",  C",  .  .  .  .  ,  etc.,     in  the  general  formulae  (159),  we  obtain 


102  THE    THEORY    AND   PRACTICE    OF   INTERPOLATION. 

F'   {t  +  n^)   =  - f a„+  (n - ^)  i„+  {f-n  +  i).„  +  (f  - f  «=+  j  I  n  - \)  d. 


+  (i'T-/+l«'-^»+iK+  ■  • 


i^"'(<  +  «o,)   =    V'-o+("-3)'^o+(f'-2«+l)''„+    .    .    .       ^  '     ^^^'^^ 


i'-'v  (<+„<„)   =  l^^(7^+(„_2)e„+    .    .    .    .^ 


/'^v  (,  +  „„)   =   Lu  + 


These  formulae  determine  the  derivatives  of  F {T)  for  any  or  all 
values  of  T  between  t  and  t-^-cj,  according  as  we  assign  different 
values  to  n.  As  in  preceding  applications,  n  is  always  a  positive 
proper  fraction. 

When,  as  is  frequently  the  case,  derivatives  are  required  for  some 
fahular  value  of  the  argument,  say  t,  we  have  only  to  make  «  =  0 
in  (164) ;    we  thus  derive  the  following  simple  exj^ressions  : 


^'"(0     =    ^('■o-S'^o+i^o-  •    • 
F^-(t)    =    ^K-2.„+    .    .    .    .) 

(t) 

F^(f)    =    4  («o-  . .   . .  ) 


(165) 


The  differences  employed  in  (IW)  and  (165)  must  be  taken 
according  to  the  schedule  on  page  3,  as  in  direct  applications  of 
Newton's  Formula. 

The  formulae  (165)  have  already  been  established  in  §18;  for  it 
will  be  observed  that  (45)  and  (165)  are  identical,  since  in  the  foi-mer 
D,D\D%.  .  .  .  are  used  symbolically  to  denote  o>F' {t),  os'F"  {t), 
w'F"'{t),.  .  .  . 


THE    THEORY   AND    I'UACTICE    OF    INTEKl'OI.ATION. 


103 


Owing  to  the  special  practical  importance    ol'  the  Jirst  derivative, 
the  coefficients  of    F'{t-\-ii(D),     namely, 


B'  =  n-\ 
C"  =  f 


/>' 


f-f«=  +  U«-i 


«  +  i 


A''  =   fl  -  i*'  +  ln--^n  + 


(1(;G) 


have  been  tabulated  in  Tal)le  IV  for  evei-y  hundredth  of  a  unit  in 
the  argument  n.  By  means  of  these  quantities,  we  readily  compute 
F'{t-\-nw)     from  the  formula 


F'{t  +  n^)   =   -  (a„+£7*„+CV„+Z)V/„+^V„) 


(167) 


The  formulae  (164),  (105),  and  (167)  are  especially  adapted  to 
the  computation  of  derivatives  at  or  near  the  beginning  of  a  tabular 
series.     We  shall  now  solve  a  few  examples  to  illustrate  their  use. 

Example  I.— From  the  following  table  of  i^(r)  =0.3^^—2  ^-+4, 
compute    F"{T)    for    T  =  2.8. 


T 

F{,T) 

J' 

J" 

J'" 

Jiv 

0 

2 

4 

6 

8 

10 

4.0 

0.8 

48.8 

320.8 

1104.8 

2804.0 

-       3.2 

+  48.0 
272.0 
784.0 

+  1G99.2 

+  51.2 
224.0 
512.0 

+  915.2 

+  172.8 

288.0 

+  403.2 

+  115.2 
+  115.2 

Here  we  have 

t  =  2 

T  =  2.8 


n   =   0.40 


a^   =    +   48.0 
i„   =    +224.0 


c,  =    +288.0 
cL  =    +115.2 


Hence,  using  the  second  equation  of  (164),  we  find 


C" 


-O.GO 


D"  =    f-  i«  +  U   =    +0.39§ 


i„  =    +224.0 
C"c^  =    -172.80 
Jj'UL  =    +   45.696 


'FJ'  =    +   96.896 


"Whence  we  obtain 


F''  =  96.890-^4 


+  24.224 


104 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


Thi.s    result   is   easily  verified  from   the    known   analytical  form  of 
the  finiction  ;    thus,  since 


we  derive 


F{T)   =  0.3T''-2T'^  +  4 
F'{T)   =   1.2rs-4T         ,         F"{_T)   =  3.GT^  -  i 


Substituting     ^=2.8    in  the  last  equation,  we  obtain 

F"(T)   =    +24.224 
as  found  above. 

Example  II.  —  From    the    table    of    the    last    example,    com^Jute 
F'(T)    for     T=zO. 

Here  Ave  employ  the  first  of  (1G5).     Making    /  =  0,    we  have 


a,   =    -3.2 


b^  =    +51.2  P^  =    +172.8  </„  =    +115.2 


AVe  therefore  obtain 

F'(t)  =  |(_3.2-5^+ifJ-yp)   =  0 

The  result  is  obviously  correct  ;    foi-  we  have 

Fi{T)  =  i.2r«-4r 

which  vanishes  for     T  =  0. 

Example  III.  — Given   the    following   table   of    F{T)  =  shi'T: 
compute     F'(T)     for     T  =  8°  36'. 


r 

F(T)E:sm'^T 

J' 

J" 

J'" 

Jiv 

Jv 

o 

4 
8 
12 
16 
20 
24 
28 

0.004866 
0.019369 
0.043227 
0.075976 
0.116978 
0.165435 
0.220404 

+  14503 
23858 
32749 
41002 
48457 

+  54969 

+  9355 
8891 
8253 
7455 

+  6512 

-464 
638 
798 

-943 

-174 

160 

-145 

+  14 
+  15 

Here  we  have 


t  = 


«  =  4°  =  -^^  =  0.069813  + 
45 


r  =  8°  36' 
36 


4x60 


0.15 


THE  THEORY  AND  PKACTICP:  OF  INTERPOLATION. 


105 


Taking  tlie  coefficients  B',  C,  U  and  E'  from  Table  TV  with 
n  =  0.15,  and  the  differences  «„,  ^>o,  (•„,  ....  from  the  given  table, 
we  find,  in  accordance  with   (167), 


a„  =    +0.023858 


B'  =    -0.35 

f>^  =    +8891 

B\  = 

— 

3111.9 

C"   =    +0.19458 

r,   =    -   638 

C'-o  = 

— 

124.1 

/;'  =   -0.12881 

(/„  =    -   160 

D'd^  = 

+ 

20.6 

E'  =    +0.09358 

"o  =    +     15 

E'e,  = 

+ 

1.4 

log  (wF>„)   =  8.314- 

•94 

■■■  "i^'„  = 

+  0.020644 

log  ft)            =   8.843937 

logii"„        =  9.470857 

•■•     -f''„   = 

+0.295704 

This  result  is  easily  verified  by  observing  that 

F'(T)   =  ^(siuJT)   =  siii2r 

which,  for     T  ^  8°  36',  becomes 

F'(T)  =   sin  17°  12'  =   0.295708 

The  former  value  is  thus  seen  to  be  very  nearly  exact. 

If  the  variation  in  F(T)  corresponding  to  an  increase  of  one 
degree  in    T  were   required   in   the  present  example,  the  i-esult  would 

be,  simply, 

F'(T)   =  0.020644-^-4   =    +0.005161 

60.  To  Comjyute  Derivatives  of  F(^T)  at  or  near  the  End  of  a 
Series. —  In  this  case  the  requisite  formulae  are  derived  from  ]S[ewton's 
Formula  for  backward  interpolation  (75),  namely. 


F^  -  na_^  +  7?6_2  -  Cc^  +  Dd^^  -  Ee_,  + 


(168) 


where  B,  C,  D,  .  .  .  .  have  the  values  given  by  (16.3),  as  before  ; 
and  where  the  differences  a_i,  h_.,,  c_3,  ....  are  taken  according 
to  the  schedule  below  : 


T 

F(r) 

J' 

J" 

Jill 

Jiv 

Jv 

t  —  5(0 
t  —  4ft) 
t  —  3o) 
If -2ft) 

fZ 

F-, 

«-5 

C-4 

d_ 

e-7 

e-5 

t   O) 

F-, 

«-2 

b^. 

C-8 

t 

Fo 

a-1 

106  THE    THEOKY    AND   PliACTICE   OF   DJTERPOLATION. 

Comparing  (1(58)   with  the  general  formula  (160),  we  have 

Therefore,  substituting  the  pi-eviously  determined  values  of 
B',  B",  .  .  .  .  ,  C,  C",  .  .  .  .  ,  etc.,  in  the  general  formulae  (161), 
we  obtain 

+  (?,-'/+ !«'-!«+ i)«-6-  ••• 


F"'(t-no>)    =   ^,  (c_,-(n-?,)  d_,  +  (f-2n+  |)  e_,-  .  .  .  .)  )    (1«'') 


F'^  (t-7iM>)    =   -Jd_^-{n~2)e_,+ 

CO      V 
F"     {t-7lw)     =     -,   ((■_5-    .    . 


Making     n^O     in  (169),  we  have 

F'     (t)     =      -     ("_i+  */'_,+  i  '■-3+  i'U+iC-5+      .      .       .      .) 

F"  (t)   =    ki/'-2+''-s+n'U+le_,  +■    •    •    •) 

^""(0      =      \(''-3+Sf?-4+l«-5+      .       .       .      .)  ^       (170) 

to 

^'"  (0   =   h  ('^-.+  2^'-5+    .    .    .    .  ) 


As  above,  we  emphasize  the  relative  importance  of  the  Jirst  deriv- 
ative in  practice :  thus,  for  brevity,  we  wi'ite  the  first  of  equations 
(169)   in  the  form 

F'{t-7iw)   =   ~(a_,-B'l>_,+  C'c_,-D'd_,+  E'e_,-  .    .    .    .)  (171) 

the   coefficients    B',  C",  D',  E'    being   taken  from   Table  IV  with   the 
argument  n. 

Formulae  (169),  (170),  and  (171)  are  particularly  useful  in  the 
computation  of  derivatives  at  or  near  the  end  of  a  series  of  functions. 


THE    TIIKORY   AND   PRACTICE   OF    INTERPOLATION. 


107 


Moreover,  when  tlie  interval  m  approaches  unity,  formulae  (109)  and 
(171)  are  convenient  for  computing  derivatives  corresponding  to  the 
argument  f -\- nco,  since  they  enable  us  to  proceed  backwards  from  the 
argument  t  -\-  w  with  the  interval  1  —  n.  We  shall  now  solve  several 
examples  to  illustrate  these  ai)phcations. 

Example  I.  —  From  the  following  ephemeris  of  the  moon's  right- 
ascension  (a),  compute  the  hourly  change  in  a  at  the  instant  Feb.  3'' 
20"  24". 


Date 
189S 

Moon's  R.A. 
a 

J' 

J" 

J'" 

Jiv 

Jv 

d       h 

Feb.  1     0 

1  12 

2  0 

2  12 

3  0 

3  12 

4  0 

h       m       8 

4  49  39.68 

5  16     0.86 

5  42  26.85 

6  8  51.58 

6  35     9.06 

7  1  13.92 
7  27     1.71 

m       s 

+  26  21.18 
26  25.99 
26  24.73 
26  17.48 
26     4.86 

+  25  47.79 

B 

+   4.81 

-  1.26 

7.25 

12.62 

-17.07 

8 

-6.07 
5.99 
5.37 

-4.45 

8 

+  0.08 

0.62 

+  0.92 

s 

+  0.54 
+  0.30 

Since    the    assigned    unit    of    time    is    1    hour,    we    have     w  =  12  ; 


hence,  letting    t  =  Feb.  4''  O'',    we  find 

4<1  Qh  Qm  _  3d  20^  24™ 


=  0.30 


which  is  the  interval  reckoned  hackwards  fi'om  t  =  Feb.  4''  0''.  De- 
noting the  quantity  sought  by  Ja,  we  then  have 

JU     =     F'(f  —  7l<ji) 

We  therefore  employ  the  formula  (171)  :  thus,  taking  the  requi- 
site differences  from  the  given  sei'ies,  and  their  coefficients  from 
Table  IV,  we  obtain 


8 

a_,   =    + 

25 

47.79 

B' 

=    -0.20 

h_„  =    -17.07 

-B'l>_.   =    - 

3.414 

C 

=    +0.07833 

c_3   =    -   4.45 

+  C"-'_„    =    - 

0.349 

u 

=    _  0.03800 

d_^  =    +  0.92 

_J/</^  =    + 

0.035 

E' 

=    +0.02009 

e_,  =    +   0.30 

+  £'._,    =    + 

0.006 

.-.  o^F'      =    + 

25 

44.07 

Whence 


The  change  in  a  for  one  miiuite  (Ji«)   is  simply 


J^a 


m 


=  2'.1445 


108 


THE   THEOKY   AND   PRACTICE    OF   INTERPOLATION. 


Example  II.  —  From  the  preceding  table  of  moon's  R.A.,  compute 
the  hourly  variation  in  J^n  for  Feb.  3'^  12'';  where,  as  above,  J^a  de.iotes 
the  change  per  minute  in  Ii.A. 

Regarding  one  hour  as  the  unit  of  time,  it  is  clear  that  the  value 
of  F"{f)  given  by  (170)  is  sixty  times  the  quantity  sought:  the  ex- 
pression for  the  required  variation  is  therefore  j.^  F"{f),  where  ^^Feb. 
3"*  12''.     Accordingly,  using  the  second  of  (170),  we  find 


Hr.  Var.  ill  /J^a,  Feb.  3"  12^ 

^   xi  (-12.62-5.37 +  |iX0.02+ 5X0.54)  = 


-0».00196 


60  '^  {\2f 

Example  III.  —  Given   the  following  values  of    F {T)  '=  logg  T 
find   F'{T)    for    T=15. 


r 

F(r)  =  iog,r 

J' 

J". 

J'" 

Jiv 

Jv 

45 

50 
55 
60 
65 

70 
75 

3.80666 
3.91202 
4.00733 
4.09434 
4.17439 
4.24850 
4.31749 

+ 10536 
9531 

8701 

8005 

7411 

+  6899 

-1005 
830 
696 
594 

-  512 

+  175 
134 
102 

+  82 

-41 

32 
-20 

+  9 
+  12 

Taking    t  =  7;"),    and  using  the  first  of  (170),  Ave  find 

Fi{f)  =  l^'(6899-a^a+-V  - '*"  + V)   =  +0.01334 

Since     F\T)  =  y,?     we    observe    that   the  true  mathematical  value  of 
the  computed  quantity  is — 

^'"(0   =    rV   =    +0.01333i 

Example  IV.  —  From    the   preceding  table  of  natural  logarithms, 
compute    F"(T)     for     T=G7. 

We  let  t  =  70,  and  proceed  by  the  second  of  (1<)9),  observing  that 

70-67 


=   0.60 


Thus  we  obtain 

C"   =   n-1    =    -0.40 

//'   =    I'^-in+ji    =   +0.197 

E"  =   f-7i^+ln  -  I    =    -0.107 


b_„   =  -0.00594 

c_3  =  +102 

-C"c^   =  +    40.8 

,/_,  =  _  32 

+  D"d_^  =    -     6.3 

e_j  =  +  9 

-£"e_5  =  +     1.0 

.-.  ^^F'l„   =  -0.00558.5 

.-.  F!L„   =  -0.00022.3 

THE    THEORY    AND    PRACTICE    OF    INTERPOLATION.  109 

The  true  value  of  this  quantity  is  — 

F"(T)   =    -  4,  =    -  TT^T-,  =    -0.00022.27  .  .  . 

Gl.  Derivatives  from  STmLiNo's  Formula.  —  When  differences 
both  preceding  and  following  the  function  Fit)  are  available,  formulae 
more  convenient  and  accurate  than  the  foregoing  may  be  employed. 
The  most  useful  and  important  of  these  are  der-ived  from  Stirling's 
Formula  of  interpolation   (104),  which  is  — 

F„  =   i^o  +  «a  +  ^''o  +  C*"  +  ^f'o  +  Ee+  .    .    .    .  (172) 

where  the  differences  are  taken  according  to  the  schedule  on  page  G2, 
a,  c,  and  e  being  the  mean  differences  defined  by  (101)  ;  and  where 
£,   C,  .  .  .  .     have  the  values 


R              "-' 

^  =    2 

^         n(u'-l)        m' 
^66 

n 
"6 

^         n=(n=-l)         n^ 
24              24 

~24 

120 

120 

n'        n 
"24  "'"30 

(173) 


Whence,  deriving  the  values  of  B' ,  B'\  .  .  .  .  ,  C,  C",  .  .  .  .  ,  etc., 
from  (173),  and  substituting  these  (with  the  above  differences)  in  the 
general  formulae  (159),  we  get 


F"  (t  +  nu,)   =   ^^  (^>,+  7ic  +(f -,V)  </„+  if-")  e  + 

F">{t  +  n.)    =   -l(c+nrf„+(f-J)e+    .    .    .    .)  ^     ^^^^^ 

F"  {t  +  nm)   =  -  ('«?„+ we  +    . 


iTv   (^  +  „,)    =    ^^,{e+  .    .    .    .) 


110         THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


Making     n  =  0     in   (174),  tlie  latter  become 


F'  (0  =  ^(.,-^c^^,e-  .    .    .    .) 
^'"(0  =  -.  (''o-A.''o+  .  .   .  .) 


(175) 


^-(0  =  -.K-. 

•       ■      •) 

<U 

•       ■) 

Again,  writing  — ?/  for  n  in   (174),  we  obtain 


F"'(t-nu>)   =   i^(^«-,«/„+(f-i)e-  .    .    .    .) 
F"  (f  —  no>)   =   -,  (d^  —  ne  +  .    .    . 


(176) 


F-   (?-«a,)   =   -^[e- 


The  coefficients  for  the  computation  of    F'(f±uco),    namely 


B'  =  n  ,         I)'  =  f-f^ 

C  -   "'  -  1  E'  =  #i  -  s'  +  J.  ^       ' 

are  given  in  Table  V  with  the  argument  n.  The  quantity  F'{T)  is 
thus  readily  computed  (for  any  value  of  T)  by  either  one  or  both  of 
the  formulae 

F'{t+nm)   =  ^{a  +  7ih^+C'r  +  D'd^  +  F'e)  (178) 

F'(t-no,)   =   '^(a-nb^+C'c-B'cl^  +  F'e)  (179) 

in  which  the  odd  differences  are  algebraic  means  of  the  tabular  differ- 
ences, taken  as  indicated  below  : 


THE    TIIEOKY    AND   rEACTICE    OF   INTERPOLATION. 


Ill 


T 

F(T) 

J' 

J" 

J"' 

Jlv 

Jv 

t  —  (1) 

^-i 

«' 

?/ 

c' 

rf' 

e< 

t 

Fo 

{a) 

^'o 

p. 

r/„ 

t  +  <o 

F. 

*1 

6?, 

The  formulae  (174)  and  (175)  may  also  be  obtained  by  the  fol- 
lowing method,  which  reverses  the  preceding  order  of  develoj^ment  by 
deriving  first  the  particular,  and  from  the  latter,  the  more  general  of 
the  two  groups  in  question. 

Expanding    F(t-\-n(ti)    by  Taylor's  Theorem,  we  have 


F{t-\-n,^)   =   F{t)-\-nu>F'{t)-]--^F"(l)Jr-—-F"'{t)^ 

11  li 


(180) 


Arranging    Stirling's    Formula     (104)     according    to    ascending 
powers  of  n,  we  find 


F(t  +  nuy)    =    /;+„,(«_!  c+^V  -  •    •    ■)+ j^(''o-iV'^o+  •    •    •) 
+ 


(181) 


Whence,  by  equating  coeflScients  of  like  powers  of  n  in  the  equivalent 
expressions   (180)   and   (181),  we  obtain 


^Fi   (0   =   a-lc  +  ^\e 
„^F'"(t)   =   c  -ie+    .    . 


(ISlrt) 


which  agree  vdth  the  formulae   (175). 

Again,  by  Taylor's  Theorem,  we  have 


F'  (t  +  nw)    =   F'  (t)  +  na,F"  (t)  +  ^  F'"  (t)  + 
F"{t  +  WO))   =   F"  (f)  +  nu,F"'(t)  +  ^  F"  (t)  + 


112 


THE   THEOKY   AND    PRACTICE    OF   INTERPOLATION. 


which  may  be  written  in  the  form 

1    /  n- 

F"(t+  nu,)   =   -U^F"(t)+  nm'F"'(f)+  —  m*F''-  (f)  + 


Substituting     in    these     equations    the    expressions    for      wi^'  (/), 
^F"{t),  .  ,  .  .  ,     as  given  by   (181«),  we  get 


F'    {t-\-  n<o)    =  - 


F"  ()•+■«<»)  =  -„ 


F>"{t.^n^)   =  -3 
F^  (t+na,)   =   -, 


)  + 


+  !L(e_  .    .)  +  .    . 

\3_ 

(c-ie+   .    .)  +  «K--    O  +  ^-Ce--    •)  + 
(d,-  .    .)  +  7i(e-  .    .)+   •    • 
(e-  .    .)+    .    . 


(182) 


These  expressions,  upon  being  arranged  according  to  the  succes- 
sive orders  of  differences,  will  be  found  identical  with  the  formulae 
(174).  For  some  purposes,  however,  the  present  foi-ni  is  more  con- 
venient. 

It  is  quite  common,  particularly  in  an  astronomical  ejihemeris,  to 
tabulate  the  values  of  F'  {T)  corresponding  to  the  tabular  values  of 
F{T).     Such  a  table  woidd  run  as  follows  :* 


T 

F(T) 

F'{T) 

t-2^ 

I'-. 

F'(t-2o,) 

t    —   CO 

^'-1 

F'(t-w) 

t 

Fo 

F\t) 

t  +  w 

l'\ 

F'(t  +  a,-) 

t  +  2io 

1'. 

F'(t+2io) 

*  It  is  evident  tliat  F'(<+H(j)  can  be  derived  from  tlie  colunm  F' (T)  by  direct  interpolation  : 
moreover,  wlien  tlie  tal)ular  values  of  F'  (T)  are  tlms  available,  this  method  of  coiiumting  F'  {l+iiu) 
is  more  expeditious  than  the  use  of  formula  (178). 


TIIK    TIIEOKY    AND   PRACTICE    OF   INTEKPOLATION. 


IV.l 


The  first  of  llio  fonniilac  (17/))  is  almost  invariably  used  for  this 
purpose,  because  of  its  simplicity  and  rapid  convergence;  this  formula 
is,  in  fact,  the  most  important  and  useful  of  those  which  pertain  to 
the  computation  of  derivatives.  For  this  I'cason  we  fornuilate  tiie 
following 

Rui.E  for  computing  the  lirst  derivative  of  a  tabular  function 
corresponding  to  one  of  the  given  functional  values  :  From  the  inean 
of  tJui  two  first  differeiices  which  immediately  pixcede  and  follow  the 
function  in  question,  subtract  one-sixth  (J)  the  mean  of  the  correspond- 
ing  third  differences,  and  divide  the  result  hij  the  tabular  interval. 
This  rule  neglects  only  5th  and  higher  diffei'cnces.  To  include  5th 
and  6th  differences,  add  to  the  above  terms  (Ixfore  dividing  by  co)  one- 
thirtieth  (g'o)  the  mean  of  the  corresjwnding  ffth  differences,  and,  divide 
by  0)  as  hefore. 

It  will  evidently  suffice,  in  most  cases,  to  apply  only  the  first  part 
of  the  above  rule. 

Several  examples  will  now  be  solved  as  an  exercise  in  the  nse  of 
the  preceding  formulae. 

Example  I.  —  Given  the  following  ephemeris  of  the  sun's  decli- 
nation (8)  :  compute  the  houi'ly  differ&ice  in  8  for  the  dates  Jan.  7, 
10,  13,  and  16. 


Date 

Sun's  Decl. 

J' 

J" 

J'" 

1  ,■ 

Diff.  for 

1S9S 

S 

6'' 

1  hour 

O           /               // 

/      // 

/      // 

// 

II 

II 

// 

Jan.  1 
4 

7 
10 

i;: 

-22  59     2.4 
22  41  38.5 
22  20  12.4 
21  54  49.4 
21  25  35.9 

+  17  23.9 
21  26.1 
25  23.0 
29  13.5 
32  56.9 
36  32.2 

+  .39  58.2 

+  4     2.2 
3  56.9 
3  50.5 
3  43.4 

-5.3 
6.4 
7.1 
8.1 

-9.3 

+  1404.55 
1638.25 
1865.20 

+  0.98 
1.12 
1.27 

+  19.52 
22.77 
25.92 

16 
19 

22 

20  52  39.0 

20  16     6.8 

-19  36     8.6 

3  35.3 
+  3  26.0 

+  2084.55 

+  1.45 

+  28.97 

The  term  3^^  e  in  the  first  of  (175)   is  hei-e  insensible  ;    hence,  for 
each  of  the  given  dates  we  have  only  to  compute  the  quantity 

Accordingly,  in  column  a  we  write  the  required  mean  first  differences, 
expressed   in    seconds    of  ai-c.     The    next   column  contains  minus  one- 


lU 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


sixth  of  the  corresponding  mean  third  differences.  Finally,  since 
w  =  72  hours,  we  Avrite  in  the  last  column  ^\^  of  the  quantities  formed 
by  suniming  the  corresponding  terms  of  the  two  preceding  columns. 
We  thus  obtain  the  hourly  diffei'cnces  required. 

Example  II.  —  Compute,  from  the  ephemeris  of  Ihe  last  example, 
the  (Idili/  motion  in  declination  Ibi-  the  date  Jan.  G''  13''  30"'. 

AVe  proceed  hacJctimrds  from  Jan.  7,  using  the  formula  (179),  and 
taking  the  coefficients  from  Table  Y  with  the  aigunient 


7' 

0' 

0"'_6"13"30"' 
3" 

10'>.5 
72'' 

=  0.14583 

find 

II 

a     =  +23'24!55 

11   = 

0.14583 

\   =  +230.9 

-7ih^    =    -       34.55 

C"  = 

-0.1560 

c    =    -     5.85 

-1  C'f    =    +    0.91 

D'   = 

-0.012 

d,   =  -  1.1 

-D'd^  =  -    0.01 

.-.  ,oF'_„  =    +22  50.90 

Whence,  for  the  daily  motion  in  S,  Jan.  (5''  13''  30'",  we  obtain 

F'_„  =  22'  50".90-:-3   =    +7'  30".97 

Example  III.  —  The  following  table  gives    F(T)  =  e%    where  e 
denotes  the  base  of  natural  logarithms:  compute  F'(T)   for   ^=0.30. 


T 

F{T)  =  cT 

J' 

J" 

J/" 

Jiv 

Jv 

0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 

1.000000 
1.105171 
1.221403 
1.349S59 
1.491825 
1.048721 
1.822119 
2.0i;^753 

+  105171 
110232 
128450 
141966 
156890 
173398 

+  191634 

+  11061 
12224 
13510 
14930 
16502 

+  18236 

+  1163 
1286 
1420 
1572 

+  1734 

+  123 
134 
152 

+  162 

+  11 

18 

+  10 

Using  the  first  of  (175),  we  lind 

i<''(0.30)   =   .^(135211-^  +  ^)    =   1.34986 

It  will    be    observed   that    our    I'csult    is    substantially  equal   to  the 
value  o{  F{T)   for    the    same   argument,    T  ^  0.30  :     this    is    required 

by  the  relation 

F{T)   =  F>{T)  =   F"{T)  =....=   e'' 


/>^  =    +0.014930 

c  =    +149G 

7,r.  =    +          927.5 

</„=    +   152 

i>"(/^  =  +           ic.c 

e  =    +      14 

ii:"e     =    -               1.6 

THE    TIIKOIJY    AND   PKACTICK    OF    INTEKPOLATION.  115 

Example  IV.— From  the  tabic  of  Example  III,  compute  F"  (T) 
for   T  =  0.462. 

Taking  t  =  0.4  and  //  =  0.(52,  we  obtain,  by  means  of  tin-  second 
of  (174), 

71.   =   0.(52 
1)"  =   f-  -jij   =    +0.1089 
Jit,   ^    n''_  n      ^    _0.115 

'  .-.  a,^FJ'  =    +0.015872.5 

.-.  /<;"      =    +1.58725 

The  trne  mathematical  value  is  — 

F".(T)   =   F(T)   =   e''  =   eo-''-  =   1.587245  .  .  . 

62.  Derivatives  from  Bessel's  Formula.  —  Other  useful  formulae, 
convenient  for  the  computation  of  tabular  derivatives,  are  those  dei'ived 
from  Bessel's  Formula  of  interpolation  (111).  The  latter  may  be 
written  in  the  form 

F,^  =   F^  +  ,w^  +  m,  +  CV,  +  Ihl  +  Ec,^-\-  .    .    .    .  (183) 

where  the  dijfferences  are  taken  as  in  the  schedule  on  page  62,  h  and 
d  being  the  mean  differences  defined  by  (106) ;  and  where  B,  C,  .  .  .  . 
have  the  following  values  : 


n(7i  —  1)  n-       n 

~         2         ~    2^  ~  2 

7i{ii  —  \)(ii  —  \)         n^       71^        11 
^  6  ^  TT  ~  4  ■•■  12 

_    (7i  +  l)n{n-l)(n-2)  _  w^        »'         n^        n 
~  24  24~"l2~2i"''l2 

_   (n  +  l)n(n—l)(n—2}(n—i)  _     n^        m*       ?r        _?j_ 
~  120  120~48"'"4S~]20 


(184) 


Deriving  from  (184)  the  values  of  B',  B",  .  .  .  .  ,  C",  G",  .  .  .  .  , 
etc.,  according  to  (154),  and  substituting  the.se  in  the  general  foi'm- 
ulae  (159),  we  obtain 


116  THE    TI1EOI5Y    AND    I'HACTICE    OF    INTKKPOLATION. 


+  (s4    —   T  2  +    -J  4  "■  1  2  cj)  ''l  +      •      • 


F'"{t  +  nu>)    =    l('.j+(M_i-)fZ+(f -;])e,+    .    .    . 

Putting    n  :=  0    in  (185),  we  get 

tt) 

P"  (i)   =  ^  (^'-i  'i  -  A  '^  +  5^4  «i  +  •    •    •    •  ) 

0)" 

jP"'(A    =    1  (r,-i, /+()*  +  .     .     .    .  ) 

(0 
(1) 

P'-  (0  =  4  (^' -  .    .    .    .  ) 


(isr.) 


(186) 


Again,  putting    n  =  I     in   (185),  we    obtain    the  following  simple 
foi'mulae: 

F'    (f+io,)     =  -    („,_,l^,^+.,3_e__    .      .      .      .    ) 

F"(t+U)   =  \(l>-^^d+   .    .    .    .) 


/""«+.U)   =   "3(q-ic,+    .    .    .    .) 

/.^'^(/+^o.)     =     1    (r/-    .      .      .      .    ) 

7''"  (Z+Jo,)    =   i(c,-  .    .     .    .) 


(187) 


wliicli   (Ictenniiic    llic   (liM'ivativcs  of  F {T)   at    points    tiiidira;/  between 
the  labular  values    of  the;  funetion.     It    is    important    to    observe  that, 


•The  coefficient  of  e^  vanishes. 


THE    TllEOUY    AND    PUACTJCE   OF   INTEKPOLATION.  117 

unless  third  (liU'e'rciicrs  ;ii'C  considci-iiblf,  a  close  <t pjtnixhuitrion  to 
F'  {t^\oi)     is  given   by  the  simple  expression 

7'"  {t\\^)  =  -^  =  ^^=^  (lS7a) 

whieh  differs  fi'oin  tiie  (wact  fornuila  only  by  tlie  omission  oC  the  small 
(piantity 

~i-h^l""r.    .    .    .) 

The  formulae  for    the    derivatives   of    F {t — iiw)    are  deduced  from 
(lll(^).     Let  us  put,  for  brevity, 

^    =  \{b,  +  l>')         ,  d    =  i(d^  +  d')  (18S) 

and   (111^/)   becomes 

F_,^  =  7-;  -  //-''  +  BJj  -  C-i  +  Dd  -  A'c'*+  ....  (189) 

Comparing  this  expression  with  the  general  formula  (KJO),  we  find 
that  a,  ^,  y,  8,  e,  .  .  .  .  ,  in  the  latter,  are  replaced  by  <i',  It,  c,  d,  r',  .... 
in  (189)  ;  hence,  observing  these  changes,  and  substituting  the  above 
determined  values  of  i>',  B",  .  .  .  .  ,  C,  C",  .  .  .  .  ,  etc.,  in  the 
formulae   (161),  we  obtain 

F'  (t-no>)  =  i  ('«'-(«-i)^+(r-i;+ ,'.)';'-(?;'-f-ij+ .5)'^ 

4.  (  II*    n'^   I      n   1     \  ^1 

T  I-.'  4  15+5  4  T50  J  '^  ■ 

F"  (t-nu,)    =   ^Ji-{>i-i)c'  +  (yr-v,-^,)d-(f-f+^^)e'i-    . 

w-  \ 

F"'(t-uo.)    =   ^(.■•-(,i-i)d  +  (f-l)e'-  .    .    .    A  )    (1^") 

F'"  (t-nm)    =   -^Jd-{n-i)e'  + 
F-  (t-n^)    =   i 


The  values  of  B',  C,  D',  and  F',  as  computed  from    the  expres 
sions 

B'  =  n  -  i  ,  D 

C"   =  1'  -  1  +  ^5  ,  F 


,,/    n"        n'  nil  \ 

~^~^"'^^''  }        (lyi) 

/    n  n      in  1  I  ^         ^ 

—    -24 1  S  T  24   ~  T55  ) 


118 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


are  given  in  Tabic  YI  witli  tlu'  arnuuK'nt  «.     By  means  of   tliese  co- 

efficients,  values    of  F\T)   are    ivadily  computed  from    either    one    of 

the  formulae 

1 


F'  (C  +  Ho,)   =   -  ((/,+  //'/,+  C  V, +  />',/  + £'V,) 

Hi 

1 


(192) 
F'  (I- 7(0))   =   -  («' - B'h  +  C V' -  D'd  +  E 'el]  (193) 

in  Avhicli  the  enni  differences  arc  vteaHs,  taken  as  indicated  l)elow  : 


T 

F(r^ 

J' 

J" 

J'" 

Jlv 

Jv 

f   —  (0 

i^-i 

// 

^' 

^/.' 

ih) 

(■' 

(r/.) 

e! 

t 

i^o 

''. 

^^0 

rtl 

{!>) 

c\ 

(^/) 

Ci 

t    +  (0 

^1 

K 

rf, 

Several  examples  Avill  now  be  solved. 

Example  I.  —  Given  the  following-  table  of  natural  sines 


T 

i^(T)EsinT 

J' 

J" 

A'l' 

Jiv 

O 

40 
42 
44 
46 
48 
50 

0.6427876 
0.6691306 
0.6946584 
0.7193398 
0.7431448 
0.7660444 

+  263430 
255278 
246814 
238050 

+  228996 

-8152 
8464 
8764 

-9054 

-312 

300 

-290 

+  12 
+  10 

Let  it  be  required  to  find  F'{T)  for    r=45°. 
Taking    t  =  14°,    we  have 


=  2°   =   ^  =  0.0349066 


«  =  t 


Hence,  using  the  first  of  (187),  we   find 
r,   =    -300; 


a,   =    +0.0246814 
12.5 


1    f    =    + 


.-.  mF\  =    +0.0246826.5 
.-.     F'    =    +0.707106 


The  true  value  of  thi.s  ([uaiitity  is  — 

F'{T)   =  cos  T  =  cos45°  =  0.707107 


THE    TUEOKY   AND   TKACTICE    ()¥    INTEKPOI.ATIOX. 


lli» 


Example  II.  —  Fi-om  tlie  preceding  table,  conipute  the  value  of 
F"{r)    for    T=U°4S'. 

We  take    /  =  44° ;    hence    )i  =  0.40.     Accordingly,  from  the  second 

of  (185),  we  obtain 

b  =    _o.0008614 
C"  =  n-i    =    -0.10  c,  =    -300  (7%  =    +  30 


Z>"  =   ^'-n_   I    =   -0.203      <Z   =    +    11 


D'Ul 


o>-F'J   =    -0.0008586 
F'J   =    -0.70465 


The  actual  value  is  — 


F"(T)   =    -sin  T 


-sin 44°  48'  =    -0.7046.'^ 


Example  III.  —  The  table  below  gives  the  "Washington  mean  time 
of  moon's  ui)per  transit  at  the  meridian  of  Washington  : 

Washington  Moon  Culminations. 


Date 

1898 

Mean  Time 
of  Transit 

J' 

J" 

J'" 

Jiv 

Mar.  22 
23 
24 
25 
26 
27 
28 

li       m 

0  15.57 

1  1.00 

1  47.29 

2  34.88 

3  23.83 

4  13.84 

5  4.24 

III 

+  45.43 
46.29 
47.59 
48.95 
50.01 

+  50.40 

in 

+  0.86 
1.30 
1.36 
1.06 

+  0.39 

in 

+  0.44 
+  0.06 
-0.30 
-0.67 

111 

-0.38 

0.36 

-0.37 

Before  proposing  an  example  fi-oni  this  ephemeris,  it  is  proper  to 
remark  that  the  tabular  function  is  the  time  of  the  moon's  arrival  at 
a  succession  of  meridians  (in  reality  one  fixed  meridian)  whose  com- 
mon difference  of  longitude  is  24  hoiu-s.  The  argument  of  the  series 
is  therefore  the  terrestrial  lomjitudc  traversed  by  the  moon,  counted 
west  from  the  Washington  mei'idian  :  the  iutercal  of  this  argument  is 
24  hours  of  longitude. 

Now,  let  IJ  denote  the  differeiuT  in.  time  of  transit  for  1  /lonr  of 
longitude.  This  quantity  is  simply  the  first  derivative  of  the  tabular 
function:  computed  for  the  instant  of  transit  at  a  meridian  I  hours 
west  of  Washington,  the  quantity  D  expresses  the  amount  by  which 
the  local  time  of  transit  at  the  meridian  /-|-1  hours  would  exceed  the 
local    time    of   transit    at   the   meridian  /  hours,  supposing    the    rate  of 


120  THE    THEORY    AND   PKACTICE    OF    IXTKUPOLATION. 

retardation  to  remain  constant  between  the  two  transits,  and  eqnal  to 
what  it  is  at  the  moment  of  the  (ii'st.  Thns,  il'  />,,  is  tlie  value  of  I) 
for  thi'  instant  of  ti-ansit  at  AV^ashington  on  Mar.  24,  tlie  loeal  time  of 
moon's  transit  at  a  station  20  minutes  west  of  Washington  is  given 
with  sufficient  precision  by  the  formula 

T   =    Mar.  '.'4''  1"  47"'.l-'9  +  ],  1\ 
Now,  by  the   first  of  equations  (18(5),  we  lind   ibr  llu'  value  of  />„, 

D^  =   F'(t)   =   ,i^-(47.o!)-if+f -^')    =   1"'.054 

Hence  the  preceding  equation  gives 

T  =  Mar.  24"  1"  47"'.94 

In  this  manner  the  local  time  of  transit  is  simply  and  accurately 
determined  for  any  number  of  stations  within  half  an  hour  of  the 
AVashington  mei'idian. 

To  find  the  local  time  of  moon's  transit  over  a  meridian  3  hours 
west  of  Washington,  on  the  2-lth  day  of  March,  we  have  only  to  in- 
terpolate the  Washington  time  of  transit  between  the  tabular  values 
for  Mar.  24  and  Mar.  25,  as  given  above,  the  interval  fi-om  the  former 
being 

Cjll 

n   =    24..   =   0-125 

Finally,  if  it  were  required  to  compute  the  local  time  of  transit 
for  several  stations  whose  longitudes  range  from  2|  to  3^  hours  west 
of  Washington,  we  should  find  the  time  for  the  3  hour  meridian  by 
direct  intei-polation,  as  explained  above.  AYe  should  also  compute 
D  =  F'{T)  for  the  same  meridian  ;  that  is,  for  ?<  =  0.125.  Then 
the  local  time  of  transit  at  any  adjacent  meridian,  whose  longitude 
from  Washington  is  3'" -|- A.""",  is  given  b}'  the  simple  formida 

^  =  ^■+60^ 
wliere  t^  is  the  time  of  transit  at  the  3  hour  meridian. 

Example  IV  —  From  the  preceding  ephemeris,  compute  the  differ- 
ence in  time  of  transit  for  1  hour   of  longitude  (D)   at  the    instant  of 


DR.  GEORGE  F.  McEWEM 

TTIK    TIIKOKY    AND    PltAC'TTrK    OF    TNTKIM'OLATTON.  121 


moon's  transit  over  tlic  iiicridian  of  San  Francisco,  Mar.  25,  1898;    the 
longitude  from  AVasliington  being  taken  as  3''   1"'  30*  =  3''. 025. 

Here  we  use  the  fbnnula  (I!*-!)  :  thus,  taking  the  coefficients  (roni 
Table  VI  (with  the  argument  n  =  8.025  -h  24  =  0.12(;04),  and  tiic  differ- 
ences from  the  given  ephemei'is,  we  obtain 


/>"  =    -0.3740 

C"  =    -f- 0.0282 
D'  =    +0.0692 


l>  =  +1.21 
c,  =  _0..30 
d  =    -0.365 


«,   =  +48.95 

III/,   =  _   O.!;-).-; 

CVj  =  -    0.008 

D'd  =  -  0.025 


.-.  oyF'„  =    +48.464 
.-.  D  =   /'''„  =  4S'".464+-24   =    +2'".019 

Example  V.  —  Use  the  above  table  of  Moon  Culminations  to 
find  the  variation  in  D  for  24  hours  of  longitude,  at  the  instant  of 
moon's  lower  transit  over  the  meridian  of  Washington,  Mar.  24,  1898. 

The  lower  transit  at  Washington  is  evidently  the  upper  transit 
over  the  meridian  12  hours  west.  Hence,  denoting  the  required  vari- 
ation by  V,  and  regarding  1  hour  of  longitude  as  the  unit,  we  find  by 
the  second  of  (187),  for  t  =  Mar.  24, 


=  Jj  (1-33  +  2\  X  0.37)   =   +0'".059 


■) 


()3.  Interpolation  of  Function,,^  hij  3Ieans  of  their  Tabular  First 
Derivatives.  —  As  already  observed,  it  frequently  happens  that  a  table 
giving  F(T)  also  contains  the  values  of  F\T)  which  correspond  to 
the  tabular  functions.  The  object  in  thus  tabulating  the  derivative  is 
to  facilitate  the  interpolation  of  intermediate  values  of  F(T).  To 
derive  the  formula  upon  which  this  method  is  based,  we  consider  the 
schedule  below,  where  the  differences  are  those  of  the  series  F'(T)  : 


T 

F(T) 

F'{T) 

IstDiff. 

2cl 

3d 

t  -2w 

t  —     w 

^-1 

fi' 

f 

t   +        (O 

Fo' 
7'V 

«-i 

A. 

(5  +  2(0 

F, 

i'V 

122  THE    TUKOKY    AND    I'ltACTICE    OF    INTERPOLATION. 

We  sliall  iissiime  Uiat  tlic  diirfreiUTS  of  1^ (T)  beyond  .I"'  may  be 
disregarded;  hence  tlie  differences  of  jP'(J')  beyond  y  may  be  neg- 
lected in   the   above    schednle.     N^ow,  by  Taylor's  Theorem,  we  have 

n^(i>'  „ ,,      wV   ,, ,,,       "■'to*   „.  ,^„., 


^        "  [3_  li 


Again,  since 


dF'  J-F'  d^Fi 

"     ~    dt    '  "      ~    df'    '  "     ~    dt'    '  ' 

we  ol)tain,  by  means  of  llie  fornnilae   (175), 


in   wliich  we  have  put,  for  brevity, 

a  =  H"'  +  "i)       ,       V  =-  Hy'  +  y.)  (I '•»<■') 

Substituting  these  expressions  for  i^„",  i^^,,'",  and  1^^'''  in  (H>1),  the 
latter  becomes 

F„   =    7';  +  ^coT-;  +  —  {a-}.,  y)  +  f3^+  y 

[2_  li  li 

which  may  be  Avritten 

/;  =  /;  +  ««.  f  a;'+  !;  a  +  f  p^  +  -,  (f -i)  y)  (197) 


By  means  of  tliis  foi-mnla  Ave  compute  F„  in  tei'ins  of  the  differences 
of  F'(T),  instead  of  tlie  differences  of  F (T)  direct,  as  in  the  usual 
formulae  of  inter])ohition. 

Substituting  — u  foi-  //   in   (1-*T),  we  have 

F_„  =  /;_  MO, (^z-;'- !;«  +  f /?„-;;,  (f-l)y)  (198) 

The  values  of 

B  =  f  ,  r=-rl,(f-l)  (199) 


THE    THEORY    AND    I'llACTICE    OF    JNTERPOLATION. 


123 


are    given    in    T;il)le  VIII  with    llu;    argiunent   n.     By  means    of   these 
coefficients  we  readily  compute 


(200) 
(201) 


The  coeHicients  in  Tal)le  VIII  are  not  extended  beyond  m^O.60, 
since  by  this  method  it  is  invariably  more  convenient  to  jjrocced  from 
the  nearest  function   F^ . 

Example.  —  From  the  American  Epherneris  for  1898  we  take  tlie 
heliocentric  longitude  of  Mercury,  together  with  tlie  dallij  motiou,  in 
longitude,  for  a  portion  of  the  month  of  October.  The  differences  of 
the  daily  motion  are  then  taken,  as  shown  below  : 


Dale 

Helioc  Long,  of 

Daily  Motion 

a 

|8 

r 

6 

1898 

MiTciir;/ 

O           1              II 

o            /            // 

/        // 

/           ff 

II 

fl 

Oct.  11 

176  51     7.8 

4     2  34.3 

-14     0.0 

12  17.5 

10  40.4 

9     9.4 

-   7  45.8 

13 
15 

184  41  59.2 
192     6  33.3 

3  48  34.3 
3  36  16.8 

+  1  42.5 
1  37.1 

—5.4 

6.1 

-7.4 

-0.7 

17 

199     8  10.6 

3  25  36.4 

1  31.0 

-1.3 

19 

205  49  59.6 

3  16  27.0 

-1-1  23.6 

21 

212  14  54.7 

3     8  41.2 

Let  it  be    required    to  find    the    heliocentric    longitude  of  Mercury 
for  the  date  Oct.  15''  U"  21'" .0. 
Here  we  have 

T  =   Oct.  15-'  14"  24"\0   =   Oct.  15''.60 
MO)  =    T  -t  =  0''.60  n  =   0.30 


Hence,  using  Table  YIII,  in  coiniection  with    (200),  we  obtain 


F^  =  192  6' 33.3 

™  =  +0.15 

B  =  +0.0150 

r  =  -0.02,"9 


Whence 


a   =    -11  28.95 
|8„   =    +   1  37.1 
-y  =    _  0     5.75 


/;'  =  +3  36  16.8 

|«  =  -  1  43.34 

B/3„  =  +  1.46 

Ty  =  +  0.14 


Sum,  1)  =    +3  34  35.06 


F..   =   F„+  ,im  .  D  =   194°  15'  18".3 


124  THE    TllKOKV    AND    ri;Af'TlCE    OF    INTEKi'OLATJON. 

Differencing  the  given  series  of  longitudes  ami  applying  Bessel's 
Formula  of  iutei-polation,   we   find 

/-;  =  194°  15'  1S".2 

()4.  ^ljq>Jlcatl<)n  of  the  Prevcdltiy  JSLihod  of  Inicrpohdion  irhen 
the  Second  Diferences  of  the  Series  F(T)  are  JSfearhj  Constant. — 
When  the  3d  and  -tth  differences  of  F {T)  arc  small  enough  to  be 
neglected,  we  may  omit  the  terms  containing  ^^  and  y  in  the  formulae 
(197)   and    (198)  :     we  therefore  obtain 

F_.,  =  /;_«,.(  a;' -^«)  (203) 

It  will  be  interesting  to  determine  the  error  of  these  apiJroximate 
formulae  as  applied  Avhen  the  3d  diffei'cnces  of  F(T)  are  apjjreciable. 
For  this  purpose  •\ve  write    (197)   in  the  form 

K  =  ^ +"-(/•;'+ ^0  +  f -/8o+(5i-T"^)"'y 

Hence,  if  we  disregard  4th  differences  of  F{T),  and  thus  neglect 
y,  it  follows  that  the  error  in  question  is  — 

c  =    ±f  «,/?„  (204) 

Now,  from   (175),  ^ve  have 


also,  from   (195), 

Whence 

and   (204)    becomes 


^■'"(0  =  ;-.  =  ^' 


F'"(t)   =   % 


«/J„   =   c   =  z/'"  (20o) 


=    ±  »    J'"  (200) 


Since  in  practice  the  maxiuuun  value  of  n  is  0.50,  it  follows  that 
the  maximum  cri-or  resulting  from  an  application  of  the  formulae  (202) 
and  (203),  when  3d  differences  of  F{2')  arc  sensible,  is  ^V--^'"-  Hence, 
even  when  third  differences  are  considerable,  these  formulae  are  suf- 
ficiently accurate  for  many  pui'|)oses. 


THE    THKOIIY    AND    I'K'AOTIOIO    OF    INTKIiPOLATION. 


125 


Thai  tlic  foniiiiliie  (202)  and  (2().'5)  are  rujorou^^J ;/  true  wlicn  the 
8d  ditfereiKHis  of  Fl^T)  are  zero  may  be  eleai'ly  shown  from  g'eo- 
metrical  eonsiderations,  as  follows  : 

The  2d  ditterences  of  F {T)  being  supposed  constant,  it  follows 
fi-om  Theorem  VI  that  the   function   is  necessai'ily  of  the   form 


F{T)  =  a^T-  +  a,T-{-  a.-. 


(207) 


N^ow,  if  in  the  accompanying  figure  we  draw  the  rectangular  co- 
ordinate axes  OT  and  OY,  and  plot  the  curve  defined  analytically  by 
(207)  (regarding  //  =  F (T)  as  the  ordinate  coi'responding  to  tlie 
abscissa  T),  it  is  evident  that  we  ol)tain  a  parahola  whose  axis  is 
parallel  to   OY. 


Let  us  now  take 

OM  =  t 
OS   =  f  +  0, 
ON  =   t  +  noj 


Whence 


MN  =  7iu) 

MP  =  F{t)  =  /; 

NQ   =   F{t  +  ni^)    =   F„ 


Draw  the  tangents  PA,  QL  ;    also,  draw  PD  \\  QL  and  PB  \\  MN. 

ilF 
Then,  denoting     ^m    l^y  ^ni  we  have 

FJ  =   tmAPB 
FJ  =  tan  DP  D 


Hence  we  find 


NA  =   MP  +  PB  tan  APB  =   F,+  iio,FJ 
NI)  =   MP  +  PB  tan  DPB   =   /l  +  nu,?\' 


It  is  therefore  evident  that  to  find  NQ  =^  F,,,  which  lies  between 
NA  and  ND,  we  must  employ  a  value  of  F'  somewhere  between  the 
values  i^„'  and  F^.  N^ow,  let  KE  be  the  ordinate  erected  at  the  mid- 
dle point  of  MN,  and  EH  the  tangent  at  E.     Then,  by  an  elementary 


12(5 


THE    TIIEOliY    AND    PK.VCTICE   OF   INTERPOLATION. 


theorem    of  the    parabola,  llie    cliord    l'(^    is    jxinillcl    to  EII,    and    we 
have,  therefore, 


NO  =   MP  +  rr.  tan  QT'li   =   F„  +  no>F! 


(208) 


which  agrees  with  the  formula   (202). 

We  have  shown  above  that  tlie  maximum  error  i)roduced  by  appl}'- 
ing  this  formuhi  when  tlie  second  diiferences  of  F (^T)  ai'c  not  constant, 
is  4V ^"'-  Hence,  unless  the  2d  differences  of  F\T)  are  considerable, 
we  may  compute  F„  by  the  following 

Rule  :  Fhid  hij  simple  inter jwlation  the  value  of  the  talmlar 
derivative  tohich  belongs  midioay  hetiveen  the  required  function  and  the 
nearest  tabular  function  (F^) ;  multiplij  this  quantity  (F!/)  hi/  the  units 
contained,  in  the  entire  interval  [T — /),  and  ajiply  the  product  to  F^. 

Example  I.  —  Given  the  following  eiihemeris  of  the  moon's  decli- 
nation  (8)  :     compute  the  value   for  the  date  July  9'  .'3"  18™.0. 


Date 

1898 

Moon's  Decl. 
S 

Diff.  for 
1  Minute 

a 

^ 

July  9     1 
9     4 
9     7 

July  9  10 

+  6      2   14.1 

6  43  39.0 

7  24  37.4 
+  8     5     8.0 

+  13.876 
13.732 
13.582 

+  13.422 

It 

-0.144 

0.150 

-0.160 

It 

—  .006 
-.010 

Here     w  =  3"  =  180'" ;    hence,  taking  t  =  July  9''  4",  we  find 

78° 


n  = 


=  0.433 


=   0.217 


180'"  •  2 

Accordingly,  the  value  of  F'  interpolated  for   half  the  interval,  or  39 
minutes,  is  — 

F'„  =  7'V  +  la  =   13".732  -  0.217  X  0".147   =  13".700 

Whence  we  obtain 

S  =   6°  43'  39".0  +  78  X  13".700   =   7°  1'  27".0 

Since  the  value  of  n  is  nearly  one-half,  we   may  interpolate  bach- 
wards  from  July  9''  7''  with  equal  facility  :    thus  we  find 

n  =  0.507         S    =  0.283 
.-.  FLn  =   13".582  +  0.283  X  0".155    =   13".626 


THE    TIIKOUV    AND    IMtACTICE    OF   INTERPOLATION. 


121 


Wliciu'c; 

8  =   7°  24'  ;57".4  -  102  x  13".(;2G   =   7°  1'  27".55 

which  substantially  agrees  with  Ihe  above  result. 

Example  U. — From  the  Ibllowing  table  of  the  moon's  horizoiilal 
parallMx    (77).  inlerpolale  the  value   for  July  10"  IG"  24'".0. 


Date 

1898 

July  10.0 
10.5 
11.0 
11.5 

Moon's  Ilor. 
Parallax 

Diff.  for 
1  Hour 

a 

56'  2g!i 
56     2.5 
55  40.7 
55  21.1 

-2.04 

1.89 
1.73 

-1.55 

+  0.15 

O.k; 

+  0.18 

Here  we  have 

T  =   July  10''  1G".40 
o)   =   12  hours 


„    =   ^  =   0.367 


t   =   July   10''   12".00 

»    =   0.183 


We  therefore  obtain 

Fi  =    -1  ".89  +0.183  X0".1G  =    -1".8G 
.-.  -K  =  5G'  2".5  -  4.4  X  l"-86   =  55'  54 ".3 

Interpolating  bachvards  from  July  11''  0'',  we  find 
,r  =  55'  40".7  +  7.G  X  1".78   =  55'  54".2 

65,  Choice  of  Fori)iulae  in  a  Given  Case.  —  When  derivatives 
are  required  to  lie  computed  at  or  near  either  the  heginning  or  the 
enxl  of  a  tabular  series,  the  formulae  derived  from  Newton's  Formula 
of  interpolation  must  necessarily  be  employed.  In  all  oilier  eases,  the 
choice  lies  between  Stirling's  and  Bessel's  forms,  and  should  be 
decided  by  the  value  of  n.  When  n  =  0,  the  formidae  (175)  are  un- 
questionably the  best.  When  n.  =  .1,  the  group  (187)  is  especially 
convenient.  As  a  general  rule,  subject  to  change  in  certain  cases,  it 
may  be  stated  that  when  n  lies  between  the  limits  0.25  and  0.75,  the 
formulae  derived  from  Bessel's  Formula  of  interpolation  will  be  found 
most  convenient :  for  other  values  of  n,  those  derived  from  Stirling's 
Formula  should  be  employed. 


128 


THE    TIIEOKY   AND   PllACTICE    OF    INTERPOLATION. 


EXAMPLES. 


1.     Given  the  follovvino:  table  of  "  Latitude  Reduction 


9 

9"-?' 

? 

V — f' 

O 

0 

5 
10 

o'  o!oo 

1  59.53 
3  55.47 

o 

15 
20 

25 

5  44.32 

7  22.80 

8  47.93 

Compute  the  variation  of  f^r  — </'  corresponding  to  a  change  of  10' 
in  ^,  for  each  of  the  tabuhii-  values  of  the  argument.  Denote  this 
variation  by  v. 

2.  From  the  preceding  table,  find  the  change  in  v  corresponding 
to  a  change  of  one  degree  in   fr,  for   gi  =  9°  30';    also  for  </  =  22°  42'. 

3.  The  table  below  contains  the  obliquity  of  the  ecliptic  (e)  for 
every  fifth  century. 


Tear.A.D. 

e 

0 

0   / 

23  41 

43.78 

500 

37 

57.97 

1000 

34 

8.07 

1500 

30 

15.43 

2000 

23  26 

21.41 

Compute  the  variation  of  e  per  century  (e')  for  the  years  750  and 
1250. 

4.  From  the  tahle  of  e  in  Exami)le  3,  find  the  variation  of  e'  per 
century,  for  the  yi'ars  0  and  2000;  e'  denoting  the  change  in  e  for  1 
century. 

5.  Given  the  logarithm  of  the  earth's  radius  vector  (log  i?)  for 
the  followins:  dates  : 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


129 


Date 

189S 

log  R 

Date 

1898 

log  R 

Dec.  15 
18 
21 

9.9930137 
9.9929025 
9.9928085 

Dec.  24 
27 
30 

9.9927353 

9.9926858 
9.9926619 

Compute  the  hourly  change  in  log  li  for  the  dates  Dec.  18''  0'', 
Dec.  22"  12",  and  Dec.  26''  17".     Denote  the  hourly  change  by  p. 

6.  From  the  preceding  ephemeris  of  log  li,  find  the  daily  vari- 
ation of  p  for  the  dates  Dec.  lo"*  0",  Dec.  24"  0",  and  Dec.  26"  10". 

7.  The  following  table  gives  the  right-ascension  of  Mercury, 
together  with  the  liourhj  difference,  for  several  alternate  day.s  of  De- 
cember, 1898  : 


Date 

1898 

R.K.ol  Mercury 

Diff.  for 
1  Hour 

Dec.  1 

h       m       8 

18     1     2.54 

+ 12^855 

3 

18  10  60.60 

11.587 

6 

18  19  28.46 

9.915 

7 

18  26  34.57 

7.749 

9 

18  31  43.19 

+   5.009 

Compute,  by  the  formulae  (200)  and  (201),  the  R.A.  of  Mercury 
for  the  dates  Dec.  1"  11"  22'".0  and  Dec.  5"  12"  30"'.0.  Check  the 
results  by  direct  interpolation  from  the  tabular  right-ascensions. 

8.     Given  the  following  ephemeris  of  the  moon's  right-ascension : 


Date 

Moon's 

Diff.  for 

1898 

Right-Ascension 

1  Minute 

il       h 

h       til       s 

s 

Apr.  8     1 

14  27  33.52 

2.4508 

8     4 

14  34  56.35 

2  4694 

8     7 

14  42  22.48 

2.4876 

8  10 

14  49  51.86 

2.5054 

By  the    process    stated    in    the    rule  of  §64,    compute    the    moon's 
R.A.  for  the  dates  Apr.  8"  3"  0"';    1"  51™;    5"  30'";    and  Apr.  8"  7"  3(5'". 


e\FT  OF. 
DR.  "■ 


G£ORGE  F. 


ClLVrTER  IV. 

OF   MECHAJ^ICAL    QUADRATURE. 

66.  We  have  shown  in  the  preceding  chapter  that  when  a  series 
of  equidistant  vahies  of  any  function  are  known,  it  is  possihle  to  com- 
pute special  values  of  the  first  and  higher  derivatives  of  that  function, 
without  regard  to  its  analytical  form.  We  shall  now  consider  the  in- 
verse problem,  namely  :  From  a  series  of  tabular  values  of  F'(T),  to 
find 

X  =  CF{T)dT 

where  the  limits  T'  and  T"  are  numerically  assigned. 

The  solution  of  this  important  problem  is  effected  by  integrating 
the  expression  for  F{t-\-nw),  as  given  by  any  one  of  the  several 
formulae  of  interpolation,  and  then  giving  to  n  the  limiting  values 
which  correspond  to  T'  and  T."  The  method  is  wholly  independent 
of  the  analytical  form  of  the  function  F(T).  It  is  therefore  of 
especial  advantage  and  importance  in  the  following  cases  : 

(a)  When  the  function  is  ainfli/ficaJh/  nnJ-nown.  This  is  the  case 
with  graphical  records  of  continuous  observations,  so  frequently  made 
in  physical  experiments  and  tests.  As  a  common  example  we  mention 
the  indicator  diagrams  of  a  steam  engine.  It  is  usually  required  to 
find  the  area  compi'ised  between  the  "pressure"  curve,  a  fixed  base 
line,  and  two  extreme  ordinates.  This  area  may  be  found,  in  the 
generality  of  cases,  by  the  method  proposed. 

(J))  When  the  function  is  analytically  hnoimi,  hut  is  non-inte- 
yrable.  Under  this  head  are  included  the  most  important  applications 
of  the  method  in  question.     For  example,  let  it  be  required  to  find 


X  = 


where  e  is  numerically  given.      We  cannot  express  the  indefinite  inte- 


THE    THEORY   AND   PRACTICE   OF   INTERPOLATION. 


131 


gral  in  finite  form.  If  e  is  sufficiently  small  (say  e  =  0.1),  we  may 
expand  (1  —  e^  siw^  T)i  in  a  series  of  ascending  powers  of  e^  s'wr  T, 
and  integrate  each  term  of  this  expansion  separately  :  a  very  few 
terms  will  then  suffice  to  compute  X  as  accurately  as  may  be  required. 
If,  however,  the  quantity  e  is  nearly  equal  to  unity  (say  e  =  0.9),  this 
series  does  not  convei'ge  with  sufficient  rapidity  for  practical  use,  and 
hence  the  method  of  expansion  fails. 

On  the  other  hand,  given  any  value  of  e  not  exceeding  unity,  we 
can  readily  tabulate  F(T)  ^  (1  —  e^  s'm^  T)-^  for  a  series  of  values 
such  as  r=20°,  21°,  28°,  ....  52°.  Having  differenced  these  values 
of  F,  it  is  then  a  simple  matter  to  compute  X  from  the  numerical 
data  thus  furnished.  In  the  nature  of  the  case,  however,  the  process 
must,  in  general,  be  an  approximative  one;  depending,  as  docs  the 
method  of  interpolation,  upon  a  limited  number  of  (usually  approxi- 
mate)  values  of  the  function  in  question. 

The  process  by  which  the  definite  integral  of  a  function  is  com- 
puted from  a  series  of  numerical  values  of  that  function,  is  called 
mecJianical  quadrature,  or  numerical  integration.  "VVe  proceed  to  de- 
velop the  formulae  which  are  commonly  employed  for  this  purpose. 

67.  Quadrature  as  Based  upon  Neavton's  Formula  of  Interpo- 
lation.—  Suppose  that  i-\-l  values  of  F(T)  have  been  tabulated  and 
differenced  as  shown  in  the  schedule  below  : 


T 

F{T) 

J' 

J" 

J'" 

Jiv 

Jv 

t 

K 

A  ' 

t    +    0) 

F. 

^0 

J.J 

Jo" 

Jo'" 

J/" 

t  +  2u, 

i\ 

J/' 

Jl 

■ 

• 

J^ , 

t  +  (i-2)c 

F.  o 

Jl 

■JIU 

//'" 

t  +  (;-l)a, 

F^, 

^',-1 

J'U 

t  +  «a) 

F, 

Let  it  be  required  to  find  fi-om  this  table  the  value  of 

B\T)(IT 


(209) 


132        THE  THEORY  AND  PRACTICE  OF  INTERPOLATION 

Since 

dT  =   wdn 


^^'e  have  ,_,  _     ,^_  \  (210) 


and  therefore 

X  =  C'F(T)dT  =   oiC'F(f  +  7io,)dn  (211) 

Now,  b}^  Newton's  Formula,  we  have 

F(t  +  no>)   =   F^  +  nJ^'  +  BJJ'  +  CJ,'"  +  DJ^''-  +    .... 

where     B,  C,  D,  .  .  .  .     denote   the    binomial    coefficients    of  the    ?«th 
order.     Multiplying  by  dn,  and  integrating,  we  obtain 

CF(t  +  )iu,)d,i  =  C{F^  +  7iJ^'  +  BJ^"+CJ^'"+  .    .    .    .)dn 
or 

fF(t  +  tia>)dn  =  nF„+  f  J^'  +  J,"  CBdn  +  J^'"  Ccdn+    .    .    .    .  +  31        (212) 

where  31  is  the  constant  of  integration.     If,  for  brevity,  we  put 

13  =  Chdn      ,       y  =     Ccdn      ,       8  =     Chd7i    ,       ....  (213) 

then,  from  the  preceding  equation,  we  derive 

£F(t  +  n^)dn   =   F^  +  iJ^'  +  f3J,"  +  yJ,'"  +  U;^  +  ....  (214) 

Whence  we  obtain,  in  succession, 

fF{t  +  no,)dn  ==  CF(f  +  w  +  7tM>)dn  =^  Fi  +  i/l,'  +  fiJ,"  +  yJ,"'+     .... 
CF(t  +  v^)d7i  =  CF(t  +  2u,  +  7iw)dn=  F.,  +  i/1„'  +  (3.'V'  +  y.V''+     .... 


(215) 


("F(t  +  jio>)d7i  =  CF(t  +  i-U  +  7>u,)dn  =  /',_i  +  i4'_i  +  ;8J;i,  +  yz/,-'i+    •     • 

Summing  the  integrals  expressed  in   (214)   and   (215),  we  find 

r=i— 1  r-i—1  r=i—l  r= i— 1 

j"F(t  +  7iu,)d7i  =  2  F,+  i  2  J,'  +  I3  ^  J/'+y  2  ^l'"+  ....  (216) 

r=0  r=0  r=0  r=0 

The    numerical    values   of    ft,  y,  8,  ...  .     (sometimes   called   the 
coefficient!^   of  quadrature)    must    now   be    determined.     These    may  be 


THE  THEORY  AND  PKACTICE  OF  INTERPOLATION.         I'Mi 

found  directly  by  integrating'  the  expressions  for  B,  C,  D,  .  .  .  .  , 
as  expanded  in  (163),  and  then  taking  the  Hniits  of  n  according  to 
(213).  But  the  following'  indirect  method  seems  preferable,  since  it 
adds  a  significance  to  the  result.     Let  us  i)ut 

Q   =r(l  +  _y)"f/«   =!"(! +»//  +  /?/+ C//  +  /y+  .    .    .    .)du  (217) 

where  j/  is  supposed  constant.     Then,  if  we  also  put 

Q'  =  fli+!/ydn 
we  shall  have 

Q'   =   1  +  i  y  +  p,/  +  y,/  +  h/  +  a/  +  t/  +   .    .    .    .  (218) 

the  coefficients  being  those  defined  in  (213). 
Again,  put 


that  is 

and  we  find 


or 


(1  +  //)"  =  z  (219) 

?ilog(l+//)   =   logs 

dz 

log(l  +  y)   .   dii   =   — 


log(l+y) 


We  therefore  obtain 

/C  C       dz  z  (1+v)" 

(1  +  M)"c??i  =  I  zdn  =  I  , — TT-i-— •  =  , — ^,   .    ^  +  const.  =  /  ,.•'  .  +  const. 
^       •"  J  Jlog(i  +  (/)       log(l  +  y)^  log(l+y)^ 

Whence 

(1+y)" 


Q'  =  S^vr 


In 


log(H-y)J 


11  =  0 


log(l+//) 


\       2"^3       4  "^5 


Expanding   the  last   expression    by  the    Binomial    Theorem,  or  by 
direct  division,  we  obtain 

Q'  =  i  +  i.'/-TVy'+5Vi/''-7¥o//'  +  Tia/-ag^l,T/+  •  •  •  •         (221) 
Whence,  comparing  (218)   and  (221),  we  find 


(222) 


li 

= 

T5 

€     = 

+  tIs 

y 

= 

+  5^ 

t  = 

£6  3 

s 



134 


THE    TIIEOEY   AND   TKAOTICE    OF   LNTEKTOLATION. 


which  are  the  numerical  values  of  the  coefficients  of  formula  (21(5). 
It  therefore  ajipeai-s  that  the  fundamental  coefficients  of  quadrature  are 
those  in  the  expansion  of    [log  (l-|-y)]~'. 

Let  us  now  regard  the  functions  F^,  F-^,  Fo_,  .  .  .  .  Fi  as  first 
differences  of  an  auxiliary  functional  series  which  Ave  shall  designate 
'F.  A  schedule  containing  the  new  series  may  be  conveniently  arranged 
as  follows  : 


T 

'F 

¥(!•) 

J' 

J" 

J'" 

'P^ 

t 

K 

IF, 

^J 

t   +   (O 

J", 

^o" 

'F„ 

•J/ 

J'" 

t  +  2u> 

'l'\ 

F., 

J/' 

'F  , 

■ 

^'.-  , 

J"' 

t  +  (l-l)w 

'Fi 

^',-1 

^',-  1 

4'i. 

t   +    1(0 

'F 

Fi 

The  value  of  'F^,  is  entirely  arbitrary.  Having  assigned  a  con- 
venient value  to  this  quantity,  the  remaining  tei-ms  in  the  series  are 
readily  formed  by  successive  additions,  thus  : 

'/'■'    =    '7^  +  F  'F    —    'F  +  F  IF .,   =    'F-  +  F- 

"VVe  shall  now  put  the  second  member  of  (21(3)  under  a  form 
more  convenient  for  computation.     By  Theorem  I,  we  have 

^K    =  K  +  F>  +  F,    +  .    . 


.    .  +  i'Vi  =    'F,  -  'F^ 
2  ^J   =  ^o'  +^i'  +  4'    +  •    •    •    •  +'^'-1  -    F,-  /; 

r=0 
r  =  ( -  1 

2    J,'l    =    JJI  +  J,'l  +  zlJ'    +    .      .       .       .     +JIU      =      JJ   -   z/„' 

r=0 
r=i— 1 

^/i;"=  4"'+Ji"'+j„"'+  ....  +j;!!i  =  J/i-J,ii 

and  hence  (216)  becomes 

£F(t  +  no.)dn  =   (iF,-iF^)  +  i{F,-F;)  +  fi(Ji-J^i) 

+  y{J,'i-J^ii)  +  S(z//"- j;")  +  e(jr-4')  +  ■ 


(223) 


(224) 


THE    THEORY    AND   rRACTICE    OF    INTEKPOLATION. 


135 


This  formula  possesses  the  disadvantage  of  involving  differences 
^.')  ^/')  ^i",  ■  ■  ■  ■  which  are  not  furnished  by  the  foregoing  schedule. 
To  o])viate  this  difficulty,  wc  proceed  as  follov^s  : 

Put 


q  =  'F^  +  h^F,  + 13 1,'  +  yj,"  +  a  /,'"  +  £.  ir  +  ii^  +  .  .  .  . 

and  (224)   may  then  be  written 

£Fit+n,.)dn   =   q  -  {'F^  +  iF^  +  l3J,'  +  yJ,"  +  UJ"+  .    .    .    .) 


(225) 


(226) 


Upon  giving  to    n,  in  formula   (75),  the  values    -\-l,  0,  — 1,  — 2, 
— 3,  — 4,  .  .  .  .  ,     successively,  we  obtain 


'^.-    ='i^m-^. 


F,    =F, 

4'   =  ^'i- 

-1  +  J'U  +  AIL,  +  ^;l,  +  JU+  ■  ■  ■  ■ 

4"  = 

J'^„  +  2J!!J,  +  3z)ti  +  ^^U+'-    ■    ■    ■ 

4'"  = 

Jill,  +  SzT'I,  +  QJU  +  .    .    .    . 

Jr  = 

AU  +  ^JU  +  .  .  .  . 

(227) 


If  these  expressions  be  substituted  in  (225),  we  shall  have  q  in 
terms  of  the  known  tabular  differences,  and  hence  obtain  the  requii'ed 
integral  from  (22()).  To  avoid  the  labor  of  numerical  reduction  inci- 
dent to  this  substitution,  we  derive  the  result  in  the  following  indirect 
manner :     Put 


log 

(1  +  x)                           ^              r-            1 

Also,  take 

u 

*     -     1-M 

and  we  have 

a;-' 

=     lL-^{y-u)     =     ?i-'  -  M° 

x" 

=     M» 

X 

=    u    (1-m)-i=    U^l,}+     m'+    M^+    «'^+    .     .      .     . 

x^ 

=    u\\-^^)-''■  =           m2+ 2w''+3j/.^+4«^+    .    .    .    . 

x' 

=   M=(l-i/,)-'=                      m3+3?<*+6h=+    .    .    .    . 

X* 

=     !«*(!-«)-'=                                             M*+4?4='+     .     .      .     . 

(228) 


(229) 


(230) 


13G  THE    THEORY    AND   PllACTICE   OF   INTERPOLATION. 

If  now,  we  substitute  these  expressions  for  «"',  .-e",  x,  of,  ...  . 
in  the  second  member  of  (228),  we  obtain  6  in  terms  of  ir\  u",  u,  ti'\  .... 
But  it  will  be  observed  that  this  operation  is  identical  in  algebraic 
form  with  the  substitution  above  proposed  with  respect  to  (227)  and 
(225) ;    for  the  0  operation  involves  the  quantities 

$■     X-',  x",  X,  x%  x\  .    .    .    .  ;       »-',  u",   11,  u\  11*,  .    .    .    .  ; 

while  the  q  oi)ei'ation  involves,  in  precisely  the  same  algebraic  relations, 
the  quantities 

„  .       IW       W       A  I       /I  II       /I  III  -IF  F       /ti  //"        //'" 

Hence  the  i-esult  for  q  will  immediately  follow  when  the  result  for  6 
has  been  derived.  But  we  may  obtain  0  as  a  function  of  u,  in  the 
form  required,  more  simply  than  by  direct  substitution  of  the  expres- 
sions (230)   in   (228).     For,  by   (229),  we  have 

1  +  .T   =   ,r^ 

1  —  M 

whence 

log(l+.r)    =    _log(l-«)  (231) 

Therefore,  by   (228),  we  find 

0  =  ]  =  -^ ^  =  u-^-irc^+pii-y,r+8a''-m^  +  iu^-  ....         (232) 

log(l+.x)  log(l  — (>) 

Accordingly,  writing  q  for  0,  'i^i+,  for  «"',  Fi  for  ti",  j'i_^  for  m,  etc., 
as  justified  by  the  pi-eceding  reasoning,  we  obtain 

q  =  'F,^,  -iF,  +  fSJ\_,  -  yJ'.L,  +  8J,^'3  -  ^^t,  +  K-^U  -  .   .    .   .  (233) 

Substituting  this  value  of  q  in  (22G),  and  grouping  hke  terms, 
we  get 

J>  (t  +  nu,)  dn   =    ('7^,+,  -  'F;)  -  k  (F,  +  i^„)  +  /?  ( J',_, -  J'„) 

-y(J,(i,+  j;')  +  8(^r3— 'o"')-'(^i-4  +  ^o'')+    •    •    •    •  (234) 

Whence,  restoring  the  values  of  /8,  7,  S,  •  .  •  .  ,  as  given  in 
(222),  and  applying  (211),  we  have 

F{T)dT  =   to  1  F{t  +  tiu,)  dn 

=  o,\i'F,^,-'F,)  -iiF,+  F,)  -  ,V  (^'.-1-^')  -  .-4  i-JiU  +  ^o") 

-  7^5  {^'A-^n  -  xtiT  (4'l4  +  ^^o")  -  « SMii  i^U-J.')  -  ....  I       (235) 


THE    TUEOKY    AND    PltACTIUE    OF    INTEKPOLATION. 


137 


AVhen  the  tabulation  of  the  function  extends  beyond  the  value 
Fi,  it  is  sometimes  more  convenient  to  employ  the  following  formula, 
easily  obtained  from   (224)  : 

=   ^\{'F,-'F,)+i,  {F,-F,)  -  ,\,  (.//-.//)  +  ,',  (^/'-z/„") 

-7V5(4"'-^„"')+TiTi(4'^— ''V')-iig|ia4^-^o^)+    .    .    .    .\  (236) 

"We  here  emphasize  the  fact  that  the  value  of  'F^  is  wholly  arbi- 
trary. 

G8.  As  an  example  in  the  use  of  formula  (285),  let  it  be  required 
to  find* 

J ■•44° 
COS  TdT 
JO" 

using  six  places  of  decimals. 

The  first  consideration  concerns  the  tabular  interval  to  be  em- 
ployed. It  is  desirable  to  tabulate  as  few  values  of  the  function  as 
ai-e  consistent  with  a  convenient  schedule  of  differences.  In  all  cases 
the  differences  should  sensibly  vanish  beyond  the  thii'd  or  fourth  order. 
Adopting  w  ^  4°  as  a  suitable  interval  in  the  j^resent  instance,  we 
obtain  the  following  table  of  F(T)  =  cos  T : 


T 

'F 

F{T)  =  cosT 

J' 

J" 

J'" 

Jiv 

0 

20 
24 
28 
32 
36 
40 
44 

0.000000 
0.939()93 
1.853238 
2.736186 
3.584234 
4.393251 
5.159295 
5.878635 

0.939693 
0.913545 
0.882948 
0.848048 
0.809017 
0.766044 
0.719340 

-26148 
30597 
34900 
39031 
42973 

-46704 

-4449 
4303 
4131 
3942 

-3731 

+  146 
172 
189 

+  211 

+  26 
17 

+22 

Taking  t  =  20°,  and  assuming  the  arbitrary  quantity  'Fg=  0,  we 
complete  the  column  'F  by  successive  additions.  Whence,  by  (235), 
we  find 

*  In  selecting  examples  of  numerical  integration  for  the  present  chapter,  we  have  in  most  cases 
chosen  for  F{T)  some  simple,  inteijrable  function,  whose  tabular  values  are  readily  taken  or  formed 
from  various  tables  in  common  use.  By  such  selection  we  gain  in  simplicity,  while  losing  little  or 
nothing  of  generality  ;  and,  moreover,  from  thus  knowing  a  priori  the  true  value  of  the  integral 
sought,  we  are  at  once  informed  as  to  the  final  accuracy  of  each  application. 


138 


THE    THEORY   AND   PRACTICE    OF   INTERPOLATION. 


0  =   6) 


F^    +F^     =    +I.(i5<)033 


A'  -A'  =  - 

20556 

J,"  +J,"  =  - 

8180 

A"'-A"'  =  + 

65 

Jjiv   +  J^iv    _     + 

48 

'F-,    -'7';        =  +5.878635 

-  i^  {F^    +  /;    )   =  -0.829516.5 

-  ^  (JJ  -J^')   =  +         1713.0 

-  3.^  (A"  +  A"  )   =  +  340.8 

-  t'/o W-  A"')   =  -  1-' 

-  Tile (4'^  +  ^^n   =  -       0^ 


log^  =  0.703392 
logo,  =  8.843937 
log  A'  =       9.547329 


sum, 

Z 

= 

+  5.051170 

CD 

= 

*    =  IE 

A' 

= 

0.352638 

Since     Ccos  TdT^  sin  T,    we  find  for  the  true  value  of  the  defi- 
nite integrral. 


A"  =   sin 44°  -sin 20° 
=   0.694658  -  0.342020 


0.352638 


If  it  be  required  to  compute 


X  =     coi 


COS  TdT 

20° 


from   the  foregoing   table,  formula  (236)  at   once   serves    the    purpose. 
Thus  we  obtain 


(i  =  2) 

'F„    -'F^       =   +1.853238 

F„     -  F^      =    -56745 

+    H^2    -  ^0    )   =    - 

28372.5 

J^'    -  J^'     =    -   8762 

-  A  (A'  -A')   =    + 

729.3 

///'  _j;'    =    +     318 

+  ^VW-^o")   =    + 

13.3 

J^n,  _  j^ni  =    +        43 

-tV\(^2"'-  A'")  =    - 

1.1 

Z  =    +1.825607 
.-.  X  =       0.127451 

Here  the  true  value  evidently  is  — 

X  =  sin  28°  -  sin  20°  =  0.127451 

69.  Precepts  for  Computing  the  Definite  Integnil  ivhen  One  or 
Botli  Limits  Fail  to  Coincide  with  some  Tabular  Yalue  of  the  Argu- 
ment  T.  —  Thus  far  we  have  considered  the  limits  of  the  integral 


X 


J'KT" 
F{T) 


dT 


to  be  of  the  form 


where  t"  and  i"  are  integers,  and  hence  T'  and   T"  are  two  jiarticular 


THE    THEORY    AND    PRACTICE   OF    rN^TERPOLATION.  139 

values  of  T  for  which  F (T)  has  been  tabuhitcd.  We  sliall  now  con- 
sider the  more  general  2'i"oblem  of  finding  X  when  the  limits  have 
the  form 

T'  =   t  +  n'u>         ,  T"   =   t  +  n"a, 

where  ii  and  n"  are  non-integers  —  that  is,  either  proper  fractions  or 
mixed  numbers. 

To  illustrate  the  significance    of  the  problem  in  question,  suppose 
it  were  required  to  find  by  mechanical  quadrature  the  value  of 


/»42''  46'  64" 

=   I  COS  TdT 

i/Sl"  13'  37" 


Obviously,  it  would  be  impracticable  to  tabulate  the  function 
for  a  series  of  equidistant  values  of  T,  of  which  T'  =  21°  13'  37" 
and  T"  =^  42°  46'  54"  are  two  particular  terms.  We  may,  however, 
employ  the  same  table  as  was  used  in  the  preceding  examples,  con- 
structed for  T  =  20°,  24°,  28°,  ....  44°,  and  obtain  the  required 
result  by  interpolation.  Thus,  in  the.  examples  jiist  mentioned,  we 
have  computed  the  values  of  X  from  the  lower  limit  T'  =  20°  to 
the  up])er  limits  T"  =  44°  and  28°,  respectively.  In  like  manner, 
keeping  the  lower  limit  always  ^  20°,  we  may  find  the  integral  cor- 
responding to  each  of  the  following  values  of  the  upper  limit,  viz.: 

T"  =  20°,  24°,  28°,  ....  44°,  respectively; 

that  is,  for  each  of  the  tabular  values  of  T.  Then,  having  differenced 
the  resulting  values  of  the  integral,  we  may  i-eadily  find  by  inter- 
polation the  values  which  correspond  to  the  upper  limits  21°  13'  37" 
and  42°  46'  54".  Denoting  these  interpolated  values  by  X'  and  X" 
respectively,  we  have 

A''=  j   cos  TdT  ,  X"  =   \  cos  TdT 

and  therefore 

/»42»   46'  64" 

X  =   I  COS  TdT 

t/Sl"  13'  87' 


A'"-  XI 


We  leave  the  detailed  solution  of  this  example  to  the  student  as 
a  valuable  exercise,  exhibiting  the  spirit  of  the  method  employed  in 
pi'oblems  of  this  type.     The  process  actually  used  differs  somewhat  in 


140 


THE    TUKOKY    AND   I'KACTICE    OF   INTERPOLATION. 


form  from    tlio    method  here    explained ;    but  the  principle  remains  the 
same.     We  proceed  to  develop  the  general  formulae. 
70.     Let  US  put 

T.  =    j  F{t  +  n(j>)dn  (237) 

and 

*(/)   =    'i<:  +  ^  F,  +  /3.7/  +  yj."  +  84'"  +  Uf  +   .   .  .  .  (238) 


where    *    denotes  an  integer.     Then  (221)  becomes 

I.  =  *(0-*(0) 


(239) 


Let  US  now  suppose  that  (239)  has   been  computed  for    i  =  0,  1, 
2,  3,  4,  ....  ,    in  succession.     Then,  from  the  sei-ies  of  values 


/„  =  *(0)-*(0) 
I,  =  *(l)-*(0) 
/„  =  *  (2)  -  *  (0) 


(240) 


thus  detei'inined,  it  is  evident  that  any  intei-mediate  value,  say  /„,  can 
be  found  by  interpolation.  To  derive  a  general  formula  foi-  this  pui'- 
pose,  we  must  express  the  differences  of  the  series  (240).  Now,  by 
(238),  we  have 


*  (0)   =    'F^  +  iF„  +  pJJ  +  yJ,"  +  8.  /,,'"  +  £J,r  + 

*  (1)    =     '/;  +  i  i-'i  +  /?,_//  +  y.  ;,"  +  8. 1,'"  +  eJ,"  + 

*(2)   =   '/;  +  ^F„+  13  J  J  +  y.JJ'  +  8.JJ"  +  cJ^^  + 


(241) 


whence,  observing  the  general  relation 

we  derive  the  following  schedule  of  differences 


Function 

1st  Differences 

2d  Differences 

3(1  Differences 

7o  =  *(0)-*(0) 
/i  =  *(l)-*(0) 
/„  =  *(2)-*(0) 
4  =  *(3)-*(0) 

/;+^j;+/?Jo"+yj„"'+. . . 

F,  +  iJ,'  +  fiJ,"  +  yJ,'"  +  .  .  . 
F„  +  \/h'  +  p/l„"  +  yJ„">  +  .  .  . 

.  In' +  i.]„"  + 13.  !„'"  +  .  .  . 

.//  +  *J,"  +  /3. /,'"  +  .  .  . 
.IJ  +  IM'  +  I3.J.,"'  +  .  .  . 

./,"+IJl"'+. . . 

Therefore,  applying  Newton's  Formula  of  interpolation,  we  have 


/„  =   /„  +  ,i(lst  ])iff.)  +  B(2d  Diff.)  +  C7(3d  Diff.)  +  . 
=   *(0) -*(()) +«(7:,+i/V  +  M"  +  7-V"+  •    • 


•) 


)  +  D(JJ>'-h  .    .)  + 


THE    THEORY   AND   TRACTICE    OF    INTERPOLATION. 


141 


By  transposing  the  term  —^(0)  to  the  first  member,  and  substi- 
tuting for  ■^(0)  in  the  second  member  the  expression  given  by  (241), 
we  find 

/„+*(0)   =   ('F„  +  iF,  +  l3J,'  +  y.J,"  +  BJ,'"+.    .    .    .) 
+  n(F,  +  iJ,'+pj„"  +  y.l,"'+  .    .    .    .) 
+  2?(Jo'  +  *^„"  +  /3.-V"+  .    .)  +  C(.J,"  +  kJ,"'-t.    .)+^W'+-    •)+•    • 

Upon  ai-ranging  the  last  expression  according  to  the  coefficients 
1,  I,  |8,  y,  8,  .  .  .  .  ,    it  becomes 

/„  +  *(0)  =  ('7^„+   nF„  +  BJ„'+CJ„"  +  I)J„'"+  .    .    .    .) 

+  i(F,+  Hzl„'  +  BJ,"+CJo"'+   .    .    .    .) 

+P(JJ+nJ„"+BJo'"+  , 

+  8(Jo"'+  . 
+  . 

Now,  it  will  be  observed  that  the  first  polynomial  in  the  second 
member  of  this  equation  is  simply  the  expression  for  'F„,  —  the  quantity 
derived  from  the  series  'F^,  'F^,  'F.;,,  ....  by  interpolation.  Simi- 
larly, the  remaining  parentheses  contain  the  expressions  for  i^„,  j'„,  //„", 
.  .  .  .  ,  likewise  derived  by  interpolation  fi'om  their  respective  series. 
AVe  therefore  have 

J„  +  *(0)   =   'F„  +  ^F,  +  li.J'„  +  y.lj'  +  h.lj"+  .    .    .    .  =   *(m)  (242) 

Whence 

CF(t  +  nu>)dn   =   J„   =   *(,i)_*(0)  (243) 

71.     In  like  manner,  if  we  put 

(r(0   =   'F,^,-^F,  +  (iJ',_,-y.JlU  +  U',1,-  .    .    .    .  (244) 

then,  by   (234),  we  have 

/,   =iiF{t+nu>)dn   =    ,y(j)_*(0) 

Therefore,  by  interpolation  (reasoning  precisely  as  above),  we 
obtain 


r^X^  +  Mw)  dn   =   (f  (n)  —  *  (0) 


(245) 


142  THE    THEOET   AND   PRACTICE    OF   LNTEKPOLATION. 

Again,  writing  n  for  the  upper  limit  n  in  (243),  and  v!'  for  n 
in   (245),  we  get 

CF{t+ni^)dn   =   *(«')-*(0)        ,  (F{t  +  nu>)dn   =   g(7i")-*(0) 

the  difference  of  which  gives 

CF(t  +  nw)dn   =   g  (»")-*  (re')  (246) 

Upon  substituting  in  equations  (243)  and  (245)  the  expressions 
for  ^  and  '/'  as  given  by  (238)  and  (244),  and  restoring  the  numeri- 
cal values  of    yS,  y,  8,  .  .  .  .    from  (222),  we  obtain 

i"F{T)dT  =   o>CF(t  +  n,D)dn 

=  <„K'i';-'i';)  +  H^;-^o)-i\(-J'„-^'«)+  .'4  i^J'-^o") 

-  t'A  (4/"-- '„'")  +  T§TJ  K^-40-^Sn5  (^-//j;  +  .    .    .    .  I         (247) 

Jf-h'tO)  /*n 

F(  T)dT  =   ,a\F{t  +  nm)  dn 

=  „K'i^„^.-'V„)-i(J^„+^„)-,->,  (J'„_,_J'„)_^i,  (4,','_2+^o") 

-7V.G^;:^3-^„"')-TtTr(^r-4+jr)-,gi3n(^i-o-^j)- •  •  •  -^  (248) 

In  like  manner,  we  derive  from   (246), 

X!+n"(0  /'  n" 

i'X  2')  rfT     =      0)1    /''((;  + «<u)rf?J 

-TVTyK'-3-4';')-ib(^';:;'-4+^;:;)-,sfirr(^;;'^-5-^nO- •  •  -i  (249) 

In  these  formulae  the  quantities  n,  n  and  n"  are  either  proper 
fractions  or  mixed  numbers;  while  the  value  of  'F^  is  wholly  arbi- 
trary. 

It  frequently  happens  that  we  have  to  compute 

X  =  fF(T)dT 


fT 
F(T) 


for  several  different  values  of  T;  the  lower  limit  remaining  fixed  and 
equal  to  t  In  such  cases  it  is  convenient  to  determine  the  arbitrai-y 
quantity  'i^^,  in  (247)  and  (248),  such  that  the  sum  of  the  terms 
having  the  subscript  zero  will  vanish.  Accordingly,  we  may  arrange 
these  formulae  as  follows  : 


THE    TIIEOKY   AND   PRACTICE    OF   INTERPOLATION, 


143 


Take 

//,'      i   A'  X      1      //  '  1      ,1  " -i-     1  il     J'" 3       //'^-i-       8  63       //v_ 

-*  0    —     —  4-'o+lI'=-'u   — 5  4  '-'o     T72u'll      — TTTS'^O    +B'iy45T7'^0  .... 

Then  — 

(a)    Tr/ze/t   the  upper   limit  falh    near   the   beginning   or 
middle  of  the  talmlar  series,  find 

F{T)dT   =    o)  j  /•'(''  +  '/ to)  r/« 

.JlflXTP  1       /'    J_     1       I   "  10      f  '"j_      M       ,/iT 86.")      //v  _L  "V 

=    "'(,  -'^i.  +  "J-f^n—  1  J-'  ,1+  54  -^'i     ~7it)''n       +  TrtlT^n         TFtJJBTT^n  T  •      •      •      -^ 

(/y)    If7?('n  the   upper  limit  falls  near  the  end  of  the  series, 
find 

F{T)dT  =   0)1  F(f  +  nm)dn 

..iiji'        4-  A' 1  y/'     '  //"  1"  ./'" ■■'   //'^  —    ««••!_  -/v    ^ 


(250) 


Example  I.  —  Let  it  be  required  to  find 


A'  = 


»0.53054 

lOdT 
Vr(i-7') 

■  0.42-37 


Here  we  adopt  the  interval  w  :=  0.02,  and  proceed  to  form  a  table 
for  T  =  0.42,  0.44,  0.46,  ....  0..54.  Instead  of  tabulating  the  given 
function,  it  is  more  expedient  to  tabulate  w  times  this  quantity.  All 
difierences  are  thus  multiplied  by  the  same  factor,  and  hence  the  final 
multiplication  by  w  is  avoided.     AVe  therefore  compute 


F{T)  =  0.02  X 


10 


0.2 


•s/T(1—T)  \JT(,1—T) 


for  the  values  of  T  given  above.     The  result  is  as  follows : 


T 

'F 

0.2 

T?IT'\   — 

J' 

il" 

J'" 

Jiv 

^(^)-  Vt(i-t) 

0.42 
0.44 
0.46 
0.48 
0.50 
0.52 
0.54 

0.000000 
0.405220 
0.808132 
1.209418 
1.609738 
2.009738 
2.410068 
2.811344 

0.405220 
0.402912 
0.401286 
0.400320 
0.400000 
0.400;i20 
0.401286 

-2308 
1626 
966 
-  320 
+  320 
+  966 

+  682 
660 
646 
640 

+  646 

oo 

14 
-  6 
+  6 

+  8 

8 

+  12 

144  THE    THEORT   AND   PRACTICE    OF   INTERPOLATION. 

The  comi^utation  is  now  readily  effected  by  foi-mula  (249).  Taking 
t  =:  0.42,  we  make  'F^  ^  0,  and  complete  the  auxiliary  series  'F. 
For  the  values  of  n'  and  n",  we  have 

0.42737-0.42 

""    = Wm =  ^-'^'^^ 

n"  =  '-'''f-'-'^^  =  5.5270  =  6  -  0.4730     ' 

Whence,  interpolating  by  Newton's  Formula,  we  obtain 

iF„,  =  +0.149G36.4  'F^,,^^  =  +2.G2 1373.8 

F„,  =  +0.404288  F,,„  =  +0.400748 

J'„,  =  -        2054  J'„,'-i  =  +  659 

z/i',  =  +  673  J':„_.,  =  +  642 

j;'/  =  _  19  z/i',;_3  =  0 

Accordingly,  by  (249),  we  find 

(iF,„^^-'F„)   =    +2.471737.4 
F„.,     +F„,      =    +0.805036  -    ^    {F„„  +  F„,)     =    -0.402518.0 

//'„,-i-^'»'     =+        2713  -   .,',  (z/'„_j-J'„,)  =    _        -226.1 

^:"-2  +^;;   =  +     isis       -  .v  {j:,,-,  +z/::)  =  -       54.8 
^:'u-^:;'  =+       19       -tV'»k'-u-</)  =  -        o-s 

.-.  A'  =    +2.068938 
To  verify  this  result,  we  observe  that 

and  therefoi'c 


A'  =  20  (sin-'  V  0.53054  -  sin"'  V 0.42737) 

=  20(168303".25- 146965".80)sinl"   =  2.068938 

Example  II.  —  Let    it    be    requiied    to    evaluate,    by    mechanical 
quadratures,  the  integrals 

A',   =  r60T''(/7'     ;     A'2  =  fGOrVr    ;     and     A',  =C(WTHT 

Here  we  tabulate  w  times  the   given  function  for    T  =^2,  4,  6,  8, 
10,  12;    thus  we  obtain  the  following  table  of  F {T)  =  120 T': 


THE    TIlKOllY    AND    PRACTICE    OE   INTEIU^OLATION. 


145 


T 

'F 

F(T)  E  1-20 r* 

J' 

J" 

Jill 

2 
4 

6 

8 

10 

12 

-       248 

+   712 

8392 

34312 

95752 

215752 

+423112 

9(50 

7080 

25920 

(51440 

120000 

207360 

+  ()720 
18240 
35520 
585(50 

+  873G0 

+  11520 
17280 
23040 

+  28800 

+  5760 

57G0 

+  57C0 

The  several  values  of  X  here  required  are  conveniently  computed 
by  the  ibrniulae  (250).  Thus  (assuming  /  ^  2)  the  first  step  is  to 
determine  'JF^,  the  computation  of  which  is  as  follows  : 


F„     =    +     9()0 

-    i  F^     =    -480 

z^'„  =  +  6720 

+  ^V^'o  =  +560 

j;'  =  +11520 

-  5'?<  =  -480 

j;"  =  +  5760 

+  t'5^-^;"  =  +152 

.-.   'i^„   =    -248 

The  column  'J^  is  now  completed    by  successive    additions   of  the 
functions  i'^,  as  shown  in  the  table  above. 
(1)   To  find  Xi  :    Here  we  have 


=   0.60 


With  tliis  value  of  n  we  readily  find  'i^„,  F'n,  /!'„,,  J'n  and  X"  by  in- 
terpolation, employing-  Newton's  Formula ;  whence  Xj  is  computed 
b}'  formula  (</)   of  (250).     The  results  are  given  below: 


F^  =  +   3932.16 

z/'„  =  +12940.8 

z/;'  =  +14976.0 

//;;'=  +  5760.0 


'F„  =  -  26.816 

+  ^  F^  =  +1966.080 

-  .fV  J'„  =  -1078.400 

+  ,ij//;'  =  +  624.000 

-iW^'n  =  -  152.000 


.:  X^   =    +1332.864 

All  of  the  quantities  above  are  mathematically  exact,  and  hence 
the  result  may  be  rigorously  comjoared  with  the  known  value  of  the 
integral :    thus,  since 


/' 


QOTHT  =  15  T* 


we  have 


A\  =  15(3.2''-2*)   =   1332.864 
which  is  identical  with  the  foregoing  result. 


14G        THE  TUEOKY  AND  PRACTICE  OF  INTEKPOLATION. 

(2)  To  find  X.:  Wc  use  tho  same  formula  as  before,  the  value 
of  n  in  this  case  being 

or    an    interval    of   0  40    counted  forward   from    the    quantities  'Fy,  F^, 
Ji,  z/,",  and  j/".     Accordingly  we  find 

'F„  =  +2461.504 

F„    =    +13271.04  +   i  F„  =  +0635.520 

J'„  =    +24460.8  -  -iV  ■^^'.,  =  -2038.400 

z/;;   =    +19584.0  +  sV /i;;  =  +  816.000 

//;;'  =    +   5760.0 -7'./iT^.',"  =  -   152.000 

.-.  A\  =  +7722.624 

This  result  is  also  mathematically  exact,  as  may  be  easily  verified. 

(3)  To  find  X3  :  Since  here  the  upper  limit  falls  near  the  end 
of  the  given  series,  we  employ  foi-mula  (h)  of  (250).  In  this  instance 
the  value  of  n  is  — 


which  corresponds  to  an  interval  of   0.20  counted  hackwards  from  the 
quantities  having  the  subscript  five.     AVe  therefore  obtain 

,t  +  1    =6   -0.20  'F„^^   =  +373075.264 

n  =5   -0.20         /''„      =    +187307.52         -    i   /';       =  -   93653.760 

n-l   =   4   -0.20         /J'„_i=    +   81139.2  -   ^V -^'^-i  =  -      0761.600 

».  -  2   =  3   -0.20         z/^La  =    +   27648.0  -   ^^  J'^^   =  -     1152.000 

n-3  =  2  -0.20         J::1s  =    +     5760.0  -  j^^  ^nU  =  -       152.000 

.-.  X^  =  +271355.904 

which  is  mathematically  exact. 

72.  (^iK  id  rata  re  us  Based,  upon  Stirling's  Formula  of  hderpo- 
lation.  —  The  preceding  formulae  of  quadrature  obviously  involve  the 
same  disadvantages  as  are  inherent  in  Newton's  Formula  of  interpo- 
lation. We  now  proceed  to  integrate  the  expression  for  F{t-\-ii(ii) 
as  given  by  Stirling's  Formula,  thus  olitaining  more  convenient  and 
accurate  formulae    than    those    already  derived.     For  this    pui'pose,   let 


THE    THEORY    AND    PRACTICE    OF    INTERPOT.ATTON. 


147 


the   schedule    of    functions   (inchiding  'F)   and  differences    be  taken  as 
below  : 


T 

'F 

F(T) 

J' 

J" 

J'" 

t  —  2a) 

F  „ 

J", 

J' , 

j"[ 

t  —  u, 

^-: 

J-: 

'F  , 

^'  ( 

J"; 

t 

F„ 

" 

^0 

'F, 

■^\ 

z/;" 

t  +  u> 

F, 

j;' 

'F, 

J', 

J'/' 

t  +  2^ 

■ 

F., 

-';' 

t  +  (i'  — l)o) 

F.  , 

4''i 

'F,  , 

J',. , 

4"\ 

t  +  iia 

F, 

z/;' 

t^-  (/+!)«) 

'^:+! 

^'Vi 

'^',+« 
^',+1 

^;;i 

^;;j 

j;;'. 

t  +  (i+2)«) 

>:,. 

^;« 

Reverting  now  to  (104),  an  inspection  of  this  equation  shows 
clearly  the  law  of  fornaation  of  the  successive  coefficients  in  the  second 
member  :    hence,  adding  the  term  in  ./",  we  have 


"2 ')  +  2"  ^°  +         6         (^        2        )  +        2r~  ^^ 

+ 120 [~^       J+  720  ^»        •    •    •    •  ^^^ 


Multiplying  by  dn  and  integrating,  we  obtain 

fF{t + wo.)  tin  =  uF^  +  f  (ju + j[)  +  s"  Ji'  +  5^5  (I*-  »i=)  (  j::; + z^:") 


(252) 


3/  being  the  constant  of  integration.  If  this  integral  is  now  taken 
between  the  limits  m  =  — |  and  ?z  =  +  i ,  the  coefficients  of 
J',  J'",  J^,  .  .  .  .     evidently  vanish,  and  we  find,  therefore, 


JF{t+  no,)  dn   ^   F,^  ^,  J„"  -  sU^  z/J7  +  ^ifi^^^  JJ' 


(253) 


(254) 


148        THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 

In  like  manner,  we  derive 

jF[t  +  nu>)d>i  =   F,+  ^\  4"  -  jijj  Jr  +  iniVVs^  ^V-  •    •    •    • 
Whence,  by  summation,  we  obtain 

J7(/  +  «a,)</«   =2/:.  +  J,^J,."-jUrr2^i'+nrfVV»wS^r-  .    .    ■    .        (255) 

Upon  substituting  the  relations 

r=i 

V  7^,  =    F^  +  F,  +  F„     +  .    .    .  +F,    =   'F.^^-'F_^ 

r  —  i 
>=0 


(256) 


in  formula   (255),  the  latter  becomes 

fFlt+  ««,) dn  =  ('F,^^-'F_{)  +  jij (j;+j - JLj)  -  ^u^ (j;;'i - j::.;) 

Finally,  therefore,  we  obtain 

F{T)dT  =   w  1  i''(i!  +  wo))(7ra 
=  o,|('i.V,_'i<'_j)+  ^V  (z^',.+,-zi'_j)-5Uij(-^.';'i-^^)+  .AV«n(/^JVi-^Ii)-  ■■■■\    (258) 

When  several  values  of  an  integral  are  to  be  computed  from  a 
given  series,  each  having  the  lower  limit  f — iw,  it  will  be  more 
convenient  and  expeditious  to  determine  the  arbitrary  quantity  '-F_^ 
such  that  the  sum  of  the  terms  with  subscript  — I  is  equal  to  zero. 
The  foi-mula    (258)   may  therefore  be  written  as  below  : 

/K"  1      //'      J.       1  7        //'"   3B7  x/v       1  \ 

-'-.J    —     — 51^^-!+  57e(J  ^-!  SJT57BB5  '^-S+     •      •     •  J 

F(T)dT  =   0,  iF{f  +  no,)dn  (259) 

1—10)  «^-l  I 

=   «»('^.+J+5'*^'.«-5H0^m+^5\VBis4''+,-    •     •      •)  / 


THE  THEORY  AND  PRACTICE  OF  INTEKPOLATION. 


1^9 


As  an  application  of  (258),  let  it  be  reqnired  to  find 

.r  =  Csec-TdT 

Taking      w  =  3°,      t  =  31°  30',       and      'F^i  =  0,       vvc    tabnlate 
F(T)  =  sec-  T    as  follows  : 


T 

'F 

F(T)  =  sec^  T 

J' 

J" 

J'" 

Jiv 

25°   30' 
28     30 
31    30 

34     30 
37     30 
40     30 
43    30 
46     30 
49     30 

0.00000 

1.37552 
2.84788 
4.43667 
6.16612 
S.06005 

1.22751 
1.29480 
1.37552 
1.47236 
1.58879 
1.72945 
1.90053 
2.11045 
2.37089 

+    6729 
8072 

9684 
11643 
14066 
17108 
20992 
+  26044 

+  1343 
1612 
1959 
2423 
3042 
3884 

+  5052 

+    2(59 

347 
404 
619 
842 
+  110S 

+  78 
117 
155 
223 

+  326 

Owing  to  the  rapid  convergence  of  the  coefficients  in  (258),  the 
effect  of  fifth  differences  is  here  insensible  :  hence,  nsing  but  three 
terms  of  this  formula,  we  obtain 


(i  =  4) 
z/',j-J'_,   =    +12920 
J'^'-J'li   =    +     899 

-'F_,  =    +8.06665 
-J'_^)  =    +        538.3 
-z?-)   =    -            2.7 

log  V  =  0.906982 
log  o>    =  8.718999 
log  X  =  9.625981 

V  =    +8.07201 

a,    =    3°   =  ,r  +  60 
.:  X  =       0.422650 

Verification :     Since 


/■ 


sec-Tdl''  =  tsLXiT 


we  have 


X  =  tan  45° 


tan  30° 


iV3 


0.422650 


To  illustrate  the  application  of  formida  (259)  when  several  values 
are  assigned  in  succession  to  the  integer  ^,  we  solve  below  an  example 
which  proceeds  according  to  the  evident  relation 


=  '.-j;:(s.)- 


DR.  GEr 


150 


THE    TIIEOKY   AND   PRACTICE    OF   INTEKPOLATION. 


where  /  denotes  the  value   of   any  coordinate  at   the  instant    T,  and  /„ 
its  value  at   the  epoch   T^.     In  particular,  let  us  put 

I  —  the  lieliooeuti-ie  longitude  of  Mars  for  any  date  T; 

—   =   the  daily  motion  in  lonuitnde  ; 
(IT  ■' 

2'^   =   1898  June  13,  Greenwich  mean  noon ; 
?    =   1°  47'  14".3   =   the  heliocentric  longitude  for  the  date  T^; 

and    let   it    be    required    to    compute    the    longitude    (I)   for    (Jreenwich 
mean  noon  of  the  dates 

(1898)  June  21,  June  29,  July  7,  July  15,  and  July  23; 

the  values  of  the  daily  motion  being  taken  from  the  American  Ei^hemeris 
for  1898. 

The  complete  solution  is  conveniently  arranged  in  tabular  form  as 

follows  : 


Date 

1898 

.(r)=s(fy 

J' 

A" 

T 

'u  +  '^ 

+£ 

I 

June    1 

9 

17 

25 

o       /         n 

5     1  36.8 
4  59  51.3 

4  57  45.4 
4  55  21.0 

-105.6 
125.9 
144.4 
160.8 

ff 

-20.4 

18.5 
16.4 

June  13 
21 
29 

0          /              // 

1  47  19.5 

6  45     4.9 

11  40  25.9 

n 

-5.2 
6.0 
6.7 

1  47  14 

6  44  59 

11  40  19 

July     3 

4  52  40.2 

175.0 

14.2 

July     7 

16  33     6.1 

7.3 

16  32  59 

11 

4  49  45.2 

187.5 

12.0 

15 

21  22  51.3 

7.8 

21  22  44 

19 

4  46  37.7 

198.1 

10.6 

23 

26     9  29.0 

-8.3 

26     9  21 

27 

4  43  19.6 

—  206.3 

—   8.2 

Aug.     4 

4  39  53.3 

The  function  tabulated  in  column  F{T)  is  eight  times  the  daily 
motion  in  I  :  it  is  so  multiplied,  because  the  unit  of  the  derivative 
being  one  day,  we  have  w  =  8  ;  and  thus  the  final  multiplication  by 
this  factor  is  avoided. 

Upon  taking  t  =  June  17,  the  formula  (259)  is  at  once  applicable. 
AVe  have,  therefore,  since  differences  beyond  J"  are  negligible. 


and 


l-L  = 


^  ilT   —    I F       J-    1    //' 


THE    TIIKORY   AND    PRACTICE   OF   INTERPOLATION.  151 

the  factor  w  having   been    previously  ai)plied.     Whence  the  expression 
for  /  becomes 

Thus,  the  value  of  /  for  ant/  date  T  being  found  by  adding  the 
constant  /„  to  the  integral  taken  from  T^  to  T,  it  is  clear  that  we 
have  merely  to  increase  the  above  value  of  'i^_,  by  the  quantity 
4  =  1°  47'  14". 3  in  order  to  avoid  the  subsequent  addition  of  this 
constant  to  each  computed  value  of  the  integral.  Accordingly,  under 
the  heading  /(,-|-'i^,  on  the  line  t — ,]  w  (=:  Jnne  13),  Ave  write  the 
quantity  1°  47'  19  ".5  ;  the  remaining  numbers  of  this  column  are  then 
formed  in  the  usual  manner  by  successive  additions  of  the  functions 
F.  Each  term  of  the  series  thus  formed  is  evidently  greater  by  Z^ 
than  if  the  latter  constant  had  been  excluded  from  the  initial  term. 

Under  -|-  ^\J'  are  written  the  values  derived  from  the  corres- 
ponding terms  in  J'.  The  sum  ^o~h -^4'2V^'  '^  then  tabulated  in 
the  final  column,  I,  which  therefore  gives  the  heliocentric  longitude  of 
Mars  for  the  dates  indicated  in  column   T. 

73.  AjypUcations  in  which  the  Limits  Fall  Otherwise  than  Midway 
Between  Tabular    Values  of  the  Argument  and  Function.  —  If  we  put 

ea+v)  =  'F,,,  +  ^\  j',„  -  jH<t  ^.'+i+  ^iiWrs  ^r+,  -  •  ■  .  .         (260) 

the  formula  (257)   becomes 

C'Fit+im) dn  =  0 (i  +  *)  -  61  (-i)  (2G1) 

Whence,  if  as  before  n  denotes  a  fractional  or  mixed  numbei', 
we  derive,  by  the  general  method  of  interpolation  employed  in  §  70, 

CF{t  +  ni^)dii   =   e{n)-e{-^)  (262) 

Upon  substituting  n  and  n"  successively  for  n,  in  (2G2),  and 
taking  the  difference  of  the  resulting  expressions,  we  get 

J^n" 
F(t  +  noy)dn  =  6{n")-e{n')  (263) 

n' 


152 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


Finally,  replacing  the  functions  6,  in  (262)  and  (2(58),  according 
to  the  expression  (2G0),  we  obtain  the  following  formulae  : 

.F(T)dT  =   o)  \F{t  +  )im)du 

=  o>\cF„-'F_o  +  ^,(^'n-^'-i)-z\hO^::'-^^)+,ifj^^(J:-Jii)-  ■  ■  -l  (264) 

J'»(-(-n"(i)  /•  n" 

F(T)dT  =  ^   \F{f  +  nw)dn 

=   «>K'^:-'-'^.0+5L(z/V'-z/'„0-jHTj(-<"-</)+f,a\¥55K''--e)-  .    .    -i    (265) 

where  the  quantity  'F-^  is  wholly  arbitrary;  and  where  'F,,,  ./'„,  j',',',  Jj,, 
....  (and  the  similar  terms  with  subscripts  u'  and  n")  arc  to  be 
found  by  interpolation. 

When  several  values  of  an  integral  are  to  be  computed  from  a 
given  series  by  (264),  the  latter  may  be  modified  to  the  more  ex- 
pedient form  given  below^ : 


F(T)dT  =   (0     F{t  +  nw)dn 

,„('F   4-     ^      /'    1'      //'"X       ■'!''''        -/' 


(266) 


•    .) 


ExAJViPLB.  —  Find  the  value  of 

X0.48 
I'-'dT 
.15 

e  being  the  base  of  the  natural  system  of  logarithms. 

Taking     w  =  0.1,  t  —  0.2,  and  F(T)  =  e%     we  prepare  the  fol- 
lowing table  : 


T 

'F 

FiT)  =  e'^ 

J' 

J" 

J'" 

Jiv 

0.0 
0.1 
0.2 
0.3 

0.4 
0.5 
0.6 

0.7 

-0.004840 

+  1.216563 

2.566422 

4.058247 

+  5.706968 

1.000000 
1.105171 
1.221403 
1.349859 
1.491825 
1.648721 
1.822119 
2.013753 

+  105171 
116232 
1 28456 
141966 
156896 
173398 

+  191634 

+  11061 
12224 
13510 
14930 

16502 
+  18236 

+  1163 
1286 
1420 
1572 

+  1734 

+  123 
134 

152 
+  162 

Proceeding  by  formula  (266),  we  find 

'F_^  =   10^  (-5'j  X  116232+ slJiyX  1163)   =    -0.004840 


THE    THEORY    AND   PRACTICE   OF    INTERPOT^ATION.  153 

whence  the  column  'F  is  completed  as  shown  above.  Denoting  the 
assigned  lower  and  upper  limits  by   T'  and   T" ,  i-espectively,  we  have 

T'  =  0.15  =  0.20-0.05  =  t-ko> 
T"  =  0.48   =   0.20  +  0.28   =   <  +  2.8  «, 

Hence,  at  the  upper  limit,  the  value  of  n  is  — 

»  =  2.8   =  2.5  +  0.30 

Accordingly,  we  find  'F,,,  .J'„,  and  ./;;'  by  interpolating  forward  from 
the  quantities  'i^„„  J'„5,  and  ./.;'^  with  the  interval  0.30.  From  the  table 
above,  we  take 

'F„,^  =    +4.058247  //'„.;  =    +0.156896  /I';;,  =   +0.001572 

Hence,  making  the  required  interpolations  by  means  of  Bessel's 
Formula,  and  jjroceeding  according  to  (266),  we  find 

'F,^  =  +4.535670.3 

z/',.  =    +0.161674  +    „\    //'„   =  +        6736.4 

j'j'  ^    +        1019  -^H^^n'  =  -  4.8 

i:  =  +4.542402 

.-.  X  =  +0.4542402 

The  true  mathematical  value  of  X  is  easily  found  :    thus,  since 

we  have 

X  =   e"-'^  -  c°'=  =  0.454240159 

74.  Quadrature  as  Based  upon  Bessel's  Formula  of  Interpo- 
lation.—  Another  set  of  formulae  for  mechanical  quadrature,  similar 
to  those  ah-eady  developed,  may  be  derived  in  the  same  manner  from 
Bessel's  expression  for  F{t-\-no}).  However,  since  these  formulae 
may  be  obtained  more  conveniently  by  a  direct  transformation  of  those 
developed  in  the  preceding  section,  we  choose  tlie  latter  course. 

Putting     n'  =  i,     and     ///  =  0,     in  formula   (263),  we  have 

f>(i  +  K«))  d)i   =6  (i)  -  e  (0)  (267) 

We  also  have,  by  (260), 

e(i)   =   'F^+^,J'.-s\lo-Jl"+^^^x>^^-  ■    •    ■    •  (2G8) 


154 


THE   TIIEOKY   AND   rilACTICE    OF   INTERPOLATION. 


Referring   now    to   tlie    general  schednle  on   page  147,  it   will   be 
observed  that  the  quantities 

are  not  explicitly  given,  but  must  be  found  by  interpolating  fo  halves 
between  'i^,_i  and  'F^+i,  j'^-i  and  J'i+i,  etc.,  respectively.  For  this 
purpose,  let  us  denote  the  algebraic  iiieans  of  the  latter  pairs  of 
quantities  by     ('F,),  (/',),  (i't"),  .  .  .  .    ;     that  is,  let  us  put 


i'F,)   =  i{'F,_,+  'F,^,) 


(269) 


Applying  formula  (126),  we  have,  therefore. 


'F,  =   i'F,)  - 

i(^'<)  +  TS^(^r)-T^,(^r)*+  •  •  ■  ■ 

^'i  = 

(J',)  -    I  (j;")  +  -ris  C'^O  -  .  .  •  . 

J["  = 

(JD  -      i  (Jj)  +  •  •  •  • 

^"i    = 

{.■11)  - .  .  .  . 

Upon  substituting  these  values  of     'Fi,  z/'j,  .J-",  .... 
and  reducing,  we  get 

e (i)  =  i'F,)  -  ^y,  (J',)+  -,y„  (J-")  -  ^hih (^0+  •  •  •  • 

Putting     i  =  0,     this  becomes 

6(0)   =   ('i^o)-TV(-^'o)  +  TVirK")-irMkK)+   •    •    ■    • 
Whence,  from  (267),  we  derive 

j"F{t+nco)dn   =   e(!)-e(0) 

+  AV  [(^A"')  -  «')]  -  TT  jUtt  [K)  -  (^:)]  +  • 


(270) 

in  (268), 

(271) 
(272) 


(273) 


*It  is  evident  from  (111)  that  the  coefficient  for  the  sixth  difference  in  Bessel's  Formula  is- 

(m+2)(M+l)  n  (n— l)(ft— 2)(n— 3) 
l!l 
which,  for     /(  =  i,     yields  the  value  given  in  the  text. 


THE    THEORY   AND   PKACTICE    OF   INTEKPOLATION.  155 

Again,  putting     n  ^  i     in   (2(32),  we  liave 

C'F{t  +  nu,)dn   =   6l(;)-6l(-i) 

=  C^)  -  Vs (./'.)  +  7V„  (./;")  -  AlU  (^0  +  .  .  .  . 

-'F_,  -  ^^J'^,+  ,\U  ^-i  -  sbWso  ^Lj+   .    .    .    .  (274) 

In  like  manner,  making     n"  ^  i-^\,     and     y/ =:  0,     in  (203),  wc 
obtain 

CF{f  +  iiiS)  dn   =   (9  (;  +  i)  -  (9  (0) 

- C^o)  +  tV  (^'o)  -  tVV  K")  +  AiU  W  -  .    .    .    .         (275) 

Finally,  substituting     u"  =  «     and     u  =z  0,     in   (2Go),    the    latter 
becomes 

n''(i!  +  Mo))(Z?t   =    e(^n)-0(O) 

—      'P     4.      1      ,/'      17         ,/"'j_  3  0  7  ,/v    _ 

-  C^o)  +  tV  (  J'o)  -  tVtT  (^-^o")  +  IfiliTT  W  -    .     .     .     .  (276) 

The   equations    (273),   (27-4),   (275)  and  (27(5)    give,   respectively, 
the  following  formulae  of  quadrature  : 

("F(T)dT  =   o>  CF(t  +  ii<o)dn 

-AUT>[(^d-Uon+  •  •  •  .  I  (277) 

J"I+<U)  /»i 

F(T)dT  =   io\  F(t  +  ni,>)d7i 

=  -  K'^.)  -  iV  (^',)  +  tVs  (4'")  -  ^JSiij  (-'O  +  •  •   •  • 

-'F_,  -  ji,  J'_,  +  sHs  ^-;  -  uirW^TJ  ^Ij  +    .    .    .    .  ^  (278) 

F{T)dT  =   wj  /'(;'+ "o,)fZw 

-  C^'o)  +  iV  (J'o)  -  ^ViT  (-^D  +  aiU  (^J)  -  ....  ^  (279) 

r/'''(7')(ZT  =   u,CF{t  +  noi)dii 

=  (0 1  '/^;+  .)j  j'„  -  ^fj^  <'  +  ^^Wbtj  ^;;  -  .  .  .  . 

-  C^o)  +  A  (^'o)  -  t'sV  C^o")  +  K-Jlit*  (^oO  -  ....  I       (280) 


156  THE   THEORY   AJSTD   PKACTICE    OF   INTERPOLATION. 

ill  which  /  denotes   an    integer   and   n   a   non-integer ;    where    'F-i    is 
wholly  arbitrary  ;   and  where    ('Fi),  {-J'i),  •  •  •  •    'i"d     {'F^),  (z/'o),  •  •  •  • 
are  vitans  of  corresponding  tabular  (juantities,  as  defined  by   (269). 
If,  in  the  formulae   (277),   (279),  and  (280),  we  take 

C^)  -  iVC''o)-rVo(X')  +  ann(^o)- •  •   •  • 
then  the  sum  of  the  terms  with  subscript  zero  will  vaiiisli.     But,  since 

the  231'eceding  condition  is  evidently  satisfied  if  we  take 

'i^_5    =    -i^o+TV(-J'o)-T'5VK")+tri^io  W-   •     •    •    •  (281) 

The   formi;lae    (277),    (278),    (279)    and    (280)    may  therefore   be 
computed  as  follows  : 

'F_,     =     -   i-Po  +  TV  (^'o)    -   tVo   i^o')  +  ,llh  (-Jo)  -    •      •      •      •  \ 

j"F'{T)dT  ^   u>CF(t  +  no,)dn  (282) 

=  -s('^:)-TV(-j'.)  +  7va-';")-ffJik(^o+  •  •  •  •  I   / 

CF'('T)dT  =   u,i'F(t+iiu,)dn  (283) 

=    <-|('^.)-TV(-^',)  +  Tyo(4"')-ITilk(^^I)+     •     •     •     •    w 

'^  =  -iJ'\+T\  u'o)  -  tVc  c^o")  +  auij  (-'o)-  ....    \ 

F(T)dT  =   <o  I  F(t +  71U,)  dn  (284) 

=  <"('/^+s+^4-^'.+j-jyu^;;i+5rfWsij^m- •  •  •  •)  ) 

rj'(r)(Zr  =  CO  rjP()'+?ia,)(Z?t  (       (285) 

,.,('FX.     1     //'     1'       /I'"  4-        3C7        //v  \  I 

—     <"  (,  ^  «  T-  ?J  ''  .1  B  -  (To  -^71     T  niTTilSlJ  ■^O   —     .      .      .      .    ;  / 

Several  examples  will  now  be  solved  as  an  exercise  in  the  use  of 
the  preceding  formulae. 

Example  T.  —  Let  it  be  required  to  find 


Y  =  Ct  si 


in  TdT 


THE    THEORY    AND    I'RACTICK    OF    INTERPOLATION. 


157 


Here  we  take       &>  =  20°  = 
F{T)  =  wTsm  T,     as  follows 


t  :=■  10°  =  :j^ ,       and  tabulate 


T 

'F 

F(T)=u>r&\uT 

J' 

A" 

J"' 

J" 

J' 

o 

-  50 
30 

-  10 
+  10 

30 

50 

70 

90 

110 

130 

+  150 

0.00000 
0.01058 
0.10197 
0.33532 
0.73607 
1.28438 

+  0.23335 
0.09139 
0.01058 
0.01058 
0.09139 
0.23335 
0.40075 
0.54831 
0.62974 
0.60671 

+  0.45693 

-14196 

-  8081 

0 

+  8081 

14196 

16740 

14756 

+  8143 

-  2303 
-14978 

+  6115 
8081 
8081 
6115 

+  2544 

-  1984 

6613 

10446 

-12675 

+  1966 
0 

-1966 
3571 
4528 
4629 
3833 

—  2229 

-1906 

1966 

1 605 

957 

-  101 

+  796 

+  1004 

0 
+  361 

648 

856 

897 

+  808 

The  value  of  X  is  now  readily  found  hj  (278).  Taking  the  arbi- 
trary quantity  'F-t^  =:  0,  we  complete  the  column  F  as  above  :  we 
then  have 

'F_,  =  J'_,  =  J'l[  =  JU  =  0 

Whence,  proceeding  by  (278),  we  find 


(i  =  4) 

('^4)  = 

=  i('F,,  +  'F,,)   =  +1.01022.5 

(J'O  =  i(J'3j  +  ^'«)  =  +0.11449.5 

-  iV  (^'4)  =  -    954.1 

{J':')=  i(J^  +  J^)   =    -   4231 

+tVVK')  =  -    64.6 

(JI)  =  ii^l,+JW    =    +        852 

-AUA^d^  -       2.7 

.-.  X  =    +1.00001 


Verification  :     Since 

C T sin  TdT  =   sin  T  -  TcosT 


we  have 


X  = 


.sin  T  -  Tcos  T 


ExAJViTLE  II. — Compute  the  value  of 


158 


THE    THEORY    AND   TRACTICE    OF    INTERPOLATION. 


Here  we  take   a>  =  0.1,  ^  =  0.9,  and  tabulate  i^(T)=(l-)-0,lT')-f, 
as  below  : 


T 

'F 

FCr)E  (1+0.1  7'2)-3 

J' 

J" 

J'" 

0.7 
0.8 
0.9 
1.0 
1.1 
1.2 
1.3 
1.4 

-0.44672 

+  0.44302 

1.30980 

2.15234 

+  2.96960 

0.93076 
0.91115 
0.88974 
0.86678 
0.84254 
0.81726 
0.79119 
0.76455 

-1961 
2141 
2296 
2424 
2528 
2607 

-26G4 

-180 

155 

128 

104 

79 

-  57 

+25 
27 
24 
25 

+  22 

Proceeding  by  means  of  (282),  we  compute  'F-^  as  follows 


F^  =  +0.88974 
(J'„)  =  -  2218 
«')  =    +  26 


-  ^  F^         =    -0.44487 

+  t'j  (^'o)    =    -        184-8 
-tVuK")   =    -  0.4 


.-.  'F_,  =    -0.44672 

Whence,  having  completed  the  column  'F,  we  conclude  the  com- 
putation by   (282),  with  the  following  results  : 

(i  =  3)  HF^)  =    +2.56097 

(jy    =    -0.02567.5  -  ^^  (jy    =    +        214.0 

(z/^'O   =    +  23.5  +7VVK")   =    +  0-4 


Z  =    +2.56311 
A'  =    +0.256311 


Since 


/< 


dT 


(1+0.1  T'')S  (1+0.1  T^ 

we  find  for  the  true  value  of  X, 

X  =  1.121936  -  0.865625  =  0.256311 


Example  III.  —  Let  it  be  required  to  find 

X 


tan  '  5 

=  fsec^  TdT 


Expressing  the  assigned  limits  in  degrees  of  arc,  they  become 


=  45° 


tan-'i]    =   56°18'35".77  =  56°.30994 


THE    THEORY    AND    PKAOTICE    OF    ISTTEKPOLATION. 


159 


We  now  take     w  ^  2°  ^  g"^  ,    t  =^  45°,     and  tabulate  the  follow- 
ing values  of     F  (T)  =  cosec*  T  : 


r 

'F 

F{T)  =  M  sec*  r 

J' 

J" 

j"i 

Jlv 

J" 

0 

41 
43 
45 
47 
49 
51 
53 
55 
57 
59 
61 

-0.00819 
+  0.07144 
0.23279 
0.42121 
0.64376 
0.90987 
1.2.3238 
+  1.62909 

0.10759 
0.12201 
0.13963 
0.1(;i35 
0.18842 
0.22255 
0.26611 
0.32251 
0.39071 
0.49608 
0.63186 

+  1442 
1762 
2172 
2707 
3413 
4356 
5640 
7420 
9937 

+  13578 

+  320 

410 

635 

706 

943 

1284 

1780 

2517 

+  3641 

+  90 
125 
171 
237 
341 
496 
737 

+  1124 

+  35 

46 

66 

104 

155 

241 

+  387 

+  11 
20 
38 
61 
86 

+  146 

Here   we   employ  formula  (285)  ;    in  which,  for   the   upper   limit, 
we  have 

n  =   (56°.30994-45°) -^2°  =   .5.65497   =   5.5  +  0.15497 


For  the  value  of  'F-^ ,  we  find 


i.;  =  +0.13963 
(J'^-)  =  +  1967 
(4',")=    +        108 


-    i  F^      =  -0.06981.5 

+    iV  (^'o)  =  +         l«'^-9 

-7VVK")=  -  1-6 

.-.  IF  ,  =  -0.06819 


Whence,  completing  the  series  'F,  and  observing  that  the  values  of 
'F„,  J'„,  and  J,',"  are  obtained  from  their  respective  series  by  interpo- 
lation with  the  interval  0.15497,  we  find 

'F„  =    +1.28846.8 
+   A    ^'»    =    +        323.1 


J\    =    +0.07754 

z/;'"   =    +        787 


_       17       /I'"     —       _ 


2.3 


.-.  X  ==    +1.29168 
Verification  :     The  expression  for  the  indefinite  integral  is  — 

Csev.'TdT  =  taur  + J  tailor 
Therefore 

with  which  the  above  result  substantially  agrees. 


160  the   theory  and  practice  op  interpolation. 

Double  Integration  by  Quadratures. 

75.  Having  derived  various  foi-mulae  for  the  mechanical  (juadra- 
ture  of  single  integrals,  the  corresjionding  formulae  for  douhle  integra- 
tion are  now  readily  deduced.  These  will  serve  to  compute  integrals 
of  the  form 

F(T)dT"-  (286) 

independently  of  the  analytical  nature  of  the  function  F{T),  provided 
T'  and  T"  are  numerically  assigned.  To  define  the  quantity  Y  more 
explicitly,  let  us  put 

('f{T)  (IT  =  /(T)  +  M  (286a) 

where  M  is  the  constant  of  integration.     We  then  have 

Y  =  CflT)aT     +M(T"-T')  (287) 

•J  T' 

It  is  therefore  evident  that  unless  the  constant  M  has  a  definite 
value  in  any  given  case,  the  value  of  Y  will  be  indeterminate.  In 
practical  applications,  however,  the  quantity  31  is  generally  known 
from  the  fact  that  the  fird  integral  has  an  assigned  value  (usually 
zero)  corresponding  to  the  lower  limit  of  integration. 

If  we  now  i)ut 

T  =    t  +  n<^         ,  T'   =    f  +  n'w         ,  T"   =   f  +  )i"w 

we  have 

(IT-    =    m-dn-  (288) 

and  hence   (28G)   becomes 

Y  =  C  CF(T)dT'  =   ,^-'  ffF(t  +  >io,)dn^  (289) 

upon  which  relation  the  subsequent  formulae  are  based. 

76.  Dotihlt  Integration  (ts  Jiased  upon  Newton's  Forrnida  of 
Interpolation. — If  we  substitute,  successively,  n  and  n'  for  n  in  (2-13), 
and  take  the  difference  of  the  two  results,  we  obtain 

CF{t-\-nio)dn  =   *(„'/)_*(„')  (290) 


tiik;  theory  and  pkactice  of  interpolation. 


101 


From   the    foni)    of  (2!)0)    it  follows   that    the    expression    for    the 
indefinite  integral  is  — 

CF{t-^nu>)d7i   =   *(«) 

or,  by  (238), 


fF(f  +  7iw)  (In  =  i^F„  (In  =  'F„+  \  F„  +  liJ'„  +  yj;,'  +  8. /;,"  +  .    .    . 


(291) 


the  constant  of  integration  being  contained  in  'F„ ,  which  depends 
upon  the  arbitrary  quantity  'F^ .  Multiplying  this  equation  by  dii,  and 
integrating,  we  get 

C  ('F{t  +  nu>)(hi-  =  C'F„dn-\-\i  F„dn-\r  pLt'„d>i.^yCr,;(hi-^oC.i;;'dn-\r  .  .   .    (292) 

Let  us  now  consider  a  new  series,  namely  — 

lip      II  w      lip       lip  up 

the  term  "F^  being  arbiti'ary,  and  the  subsequent  terms  so  determined 
that  the  qnantities 

'F      'F      'F  'F 

^0'       ^1^       ■'^2)    •       •       •       •      -f,-+I 

are  the  successive  first  differences  of  the  proposed  series.  The  man- 
ner of  arranging  the  series  "F,  'F,  and  F,  together  with  the  differences 
of  F,  is  shown  in  the  schedule  below  : 


T 

iiF 

'F 

F{T) 

J' 

J" 

J'" 

Jlv 

"F^ 

0 

'K 

f 

"F, 

0 

'F, 

K 

^\ 

t    +    O) 

"Fn 

K 

0 

■^0 

" 

'F., 

^'1 

<' 

/'  +  2w 

"/'; 

- 

^2 

•^'2 

^r 

//;" 

J'v 

t  +  3o) 

"F, 

F, 

^2' 

z/r 

t  +  (/-2)a) 

■  "^,-1 

'^'',-1 

F,-. 

^'-2 

^j;:. 

/^;-3 

'^;i4 

t   +    (j-l)o) 

"F, 

^,-1 

^,-2 

'/', 

^',-1 

t  +    /u) 

lip 
lip 

ip 

F, 

162 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


Now,  since  tlie  differences  ^"^'  may  be  regarded  as  a  series  of 
functions  whose  1st,  2d,  ....  differences  are  //»'+",  J'-'+-'>  .  .  .  .  , 
it  is  clear  tliat  formula  (291)  may  be  applied  successively  to  each  of 
the  integrals  in  the  second  member  of  (292).     Accordingly,  we  have 


dn 
(In 


I3fj'„dn 

bC/i::'  dn 

£  Cj'J  dn 


dn   = 


(293) 


Summing  these  expressions,  we  find,  in  accordance  with   (292), 


f  CF{t  +  nu>)  dn''  =   "F„+  'F„  +  (1  +  2/3)  F,^  +  (/?+2y)  //'„ 

+  {p-"  +  y  +  28)  Jl'  +  (2^y  +  8 + 2£)  J',l '  +  . 


(294) 


Upon  substituting   the   numerical  values   of     ;8,  y,  S,  .  .  .  .     from 
(222),  formula  (291)   becomes 


^  ^F{t+n>^)  div  =  "F^  +  '/'„  +  tV  i^„  -  ^iT  ^;,'  +  -k  ^:."  - 


(294a) 


the  coefficient  of  j'„  reducing  to  zero.  We  proceed  to  determine  the 
expansion  to  Avhieh  the  coefficients  of  this  formula  belong.  For 
brevity,  let  us  write   (291)   in  the  form 


rp-' ('!  +  «'-)  ''"•'  =    "^n  +  'F^  +  aF„  +  hJ\  +  cA'i  +  dJ'::  + 

Now,  from  (228),  we  have 
Also,  let  us  put 

w  =  X--  +  .(,-•  +  ax"  +  h.r  +  ,'.,■■  +  d.r^  +   .    .     .    . 


(295) 


(296) 


(297) 


THE    THEORY    AND    PRACTICE    OF    INTERPOI.ATION. 


Ki:} 


in  which  the  coefficients  are  taken  as  in  (29.3).  Whence,  since  the 
second  member  of  (295)  is  tiie  combined  sum  ol"  tlie  second  members 
in  (293),  it  is  evident  tliat  (297)  may  be  resolved,  conversely,  as 
follows  : 

IV   =  a;-2  +  I  a;-'  +  fix"  -\-  yx  +  hx-  ■\-    . 

+  i(a-'+ix''+/8«+y.«'+    .    .    .    .) 

+  y(x+  ia:-4-    . 
+    . 

which  may  be  written 


w   =        a;-i  (a;-i  +  k  x"  +  fix  +  yx-  +  &x^+    .    .    .    .  ) 
+  ix''(x-^+ix''  + I3x+ yx-+8x''i.    .    .    .    .) 

+  fix  (X-'  +lx''+fix+  yx'  +  8x'+  .  .  .  .  ) 
+  yx'  (X-'  +  ixO  +  fix+  yx-  +  8.r^+  .  .  .  .  ) 
+  8.--^  (a;-i  +  i  x"  +  fix  +  y.<--^  +  Sx'+  .  .  .  .  ) 
+ 

=    (,*->  +ix°+  fir    +  y.'-  +      .     .     .     .  ){x-'  +  *.'•"+  fix  +  yx'  + 
=    (,;-i+^,,.o+/J,,.    +y,r-^+8,,3+    .     .     .    .y 

Therefore,  by  (296),  we  have 


=     ^log(l  +  .r) 

_     r~-  4-  1 — 1  -I 1     r°  1       ;■-  -U      i      r^  '-SI      r*  J-      19      v^ 

—     ^        -f-  u,       -r    15  ■<  240  •        r  5^40  .*-  ir(T4  KO  >     ITOT^  •'' 


.»■•      X'      a;'      .?• 
=   '•^•-2  +  -3-4+5 


(298) 


Comparing  (297)  and  (298),  it  follows  that  the  coefficients  of  the 
former,  and  hence,  also,  those  of  (295),  are  the  coefficients  in  the 
expansion  of  [log  (1 -(-.*■)  ]~^,  as  developed  in  (298).  "Whence,  in- 
troducing these  valnes  of    a,  b,  c,  d,  .  .  .  .     in   (295),  Ave  obtain 

ffF(t  +  no,)dn:'="F„+  'F„+  ^^^  F,-^x^  j;'+  ^i,  ^'-^ho  '^u  +  T;iU  ^l----     (299) 

as  was  found  directly  —  in  part  —  in   (294^/). 
Let  us  now  j^ut 


X(»,)    =    "F„  +  IF,,  +  aF„  +  hJ'„  +  cJ',;  +  d.J',l'  +  eJ';;  +  .    .    .    . 

—         ■'^n-r     ^  n'T  15  -'^  n  -r  '-'-'  H  —    54(1  ^n    +  Jiff  ~'  ,i      —  SC4S5  ^"      r 


and  (299)  becomes 


C  CF{t  +  n,^)dn-   =   A  («) 


(.300) 


(301) 


l(Ji 


THE    THEORY   AND   PRACTICE    OF   INTERPOLATION. 


Whence,  if  the  intcg-ral  be  taken  between  the  two  fractional 
limits,  n    and  n",  we  shall  have 

f  pV +  »<")''«'  =  ^("")  -  >•(«•')  ('"^02) 

And  if  we  make  the  upper  limit  an  integer,  say     n"  =  i,     we  have 

r  C'f (f  +  ?! «,)  dn'   =  X(!)  -\  (w ')  (303) 

The  last  formula  involves  the  disadvantage  of  employing  differ- 
ences z/,',  Ji",  Ji'",  ....  which  are  not  given  when  the  tabulation  of 
F {T)  ends  with  the  quantity  Fi-  To  remedy  this  defect,  we  i^ro- 
ceed  as  follows  :     Put 

V   =   A(t)    =    "Fi+  'F.  +  nF,  +  h.l',  +  C.I''  +  dJ'."  +  eJY  +    ....  (304) 

and  sul)stitute  for     "i^, ,  'Fi,  F^,  z//,  j/',  ....     the  expressions 

'IF,  ="7^.+„  -  2'i^,.+i  +  F, 
'F    --  'F      —  F 


F,  =  F- 

j.'  =  j;_j  +  ,/;i,  +  j'^  +   z7;i,  +  . 

J''  =  j;i,  +  2j;:;3  +  3J!i,  +  . 

j;"  =  •          J,::;  +  3j;i,  +  . 

JY  =  J;i4  +  . 


(305) 


AVlience  the  integral  (303)   may  at  once  be  expressed  in  terms  of  the 
available  differences,     //'{_, ,  /I'i^  ?  ^'iU  > 
substitution,  let  us  put,  as  in  (229), 


.  .     However,  to  avoid  direct 


(.30G) 


and  we  shall  have 


x-^ 

= 

ic-  (1  —  ?(.)- 

= 

U--  -  2 (/-I  +  ?(» 

.T-' 

= 

H-'(l-!0 

-- 

«-i  -  »° 

x" 

= 

n" 

X 

= 

«(!-?<)-' 

= 

»,   +    „■'   +    !!»  +    «^   +     .       .       .      . 

,r2 

^ 

„2(l_„)-2 

= 

ir+2u''  +  Zu*+  .     .     .     . 

a« 

= 

u^(l-ii)-^ 

= 

m»+-3j!^+  .    .    .    . 

.r^ 

= 

u\l-v)-^ 

^ 

(^^-1-  .    .    .    . 

(307) 


TIIK    TIIKOIIY    AND    rUACTICE    OF    INTERPOLATION.  Hyi) 

Again,  from   (297),  we  liave 

w   =   X--  +  .?-'  +  a.,-"  +  bx  +  c.r-  +  J.r^  +  e.i-*+  ....  (308) 

Now,  it  is  evidcMit  tlial  if  the  expressions  (307)  he  substituted 
in  the  second  member  of  (308),  the  algebraic  process  will  he  identi- 
cal in  form  with  that  of  substituting  the  expressions  (305)  in  (;}04). 
The  w  operation  involves  the  quantities 

w  ;  x~%  x~\  x",  X,  X-,  ,/•",     ....       ;      tr^  u~\  u",   a,   u";  u^,     .    .     .    .     ; 

while  the   v  operation  involves,  in  exactly  the  same  manner,  the  (juan- 
tities 


V   •    "F      'F      F      A'      A"    A'"  •    "F  <F  F      A'  'I"         /'" 


Hence,  if  we  perform  the  w  operation,  the  residt  for  r  is  at  once 
known.  But  the  expression  which  results  from  substituting  (307)  in 
(308)  is  obtained  with  greater  exjiedition  by  the  following  process  : 
From   (298),  we  have 

Whence,  by   (306),  we  find 

w  =    |_log(l-«)|-=  =    nog(l-«)|-2 

the  expansion  of  which  is  immediately  obtained  by  writing  — a  for  ./; 
in  the  second  member  of  (297).     Thus  Ave  find 

w  =   u~''  —  11""^  +  ait"  —  hu  +  cu-  —  dii^  +  eu*  —    .    .    .    .  (309) 

Therefore,  according  to  the  pi-eceding  reasoning,  the  expression  for 
V  is  — 

V   =   >'F,^  -  '7'%,  +  aF,  -  h.J',_,  +  cj;i^  -  dJlll,  +  eJ^  -  .    .    .    . 

Denoting   this   expression   by   ■n-(')j    '^^^^^    restoring   the   numerical 
values  of    a,  b,  c,  .  .  .  .     from  (300),  we  have 

V  =  ■n-{i)   =    "F,^„  -  'F,^,  +  aF,  -  hJ\_,  +  cj;i„  -  rfj;i'3  +  eJ'I^  _  .    .    .    . 

=    "i^+,  -  'i^+i  +i,F,-  ^_U  ^'U  -  ^.U  ^;-3  -  ,T*Ho  41.  -  ...    .  (310) 


IGG  TTIK    TIIEOKY    AND   ntACTTrE    OF    INTERPOLATION. 

Whence,  by   (304)   and   (310), 

X  (i)    =    V    =    IT  (i) 

and  tlu'  fonmda    (303)   becomes,  therefore, 

C  ^'F(t  +  nm)dn''  =   Trii) -\{n')  ■  (311) 

In  the  fonmda  just  proved  tlie  quantity  /  denotes  an  intcg-er. 
Now,  by  the  general  method  of  interpolation  employed  in  §70,  it  is 
easily  shown  tliat  (311)  is  true  for  non-integral  valnes  of  i.  Thus, 
writing  n"  for  /,  this  formula  becomes 

r  rF{t  +  ««,) (hi"  =  TT (n")  -  X  (■»')  (312) 

We  now  bring  together  equations  (300),  (310),  (302),  (312) 
and  (289),  in  the  order  named  ;  observing  that  in  the  first  two  of 
these  we  may  write  "i^„_^,  for  "jP„-(-'i^„  and  for  "F,,.^^  —  '-^«+i  >  respec- 
tively.    Thus  we  obtain  the  following  group  : 

\(n)  =  "F„^,+  j\,F„-,uj':  +^i„j;"-  ,Hif.^r  +ai?^:  -  .  .  . 
^(n)  =  "F„^,  +  ^^  f„-^i^j:u-^u^::i,-  ,m^  j^,-^iuj:_,-  .  .  . 

C  ("F{t  +  «<o)  d7i'   =   X  («")  -  X  (n') 
C  CF(t  +  Hu)  dn-   =   TT  {II ")  —  X  (Ji') 

/  /-•    /'l+n"<Jl  „      /*    /•>'" 

F  =  |    \F(T)dT-'        =   (0^       i  F(t  +  iio,)d>r 

From  this  group  are  immediately  derived  all  of  the  formulae  given 
in  the  following  section. 

77.  We  have  already  remarked  that  in  tlie  jirocess  of  single  in- 
tegration the  value  of  the  definite  integral  is  wholly  independent  of 
the  absolute  value  of 'F^,  which  may  therefore  be  assigned  arbitrarily. 
Similarly,  in  double  integration,  the  quantity  "F^  may  be  taken  at 
pleasure,  the  integral  being  independent  of  its  absolute  value.  Per 
contra,  the  douhlt  integral  will  evidently  vary  with  the  value  assigned 
to  'F^.  Hence,  unless  'F^  is  fixed  by  some  special  considei-ation,  the 
value  of  the  double  integral  is  indeterminate  —  a  conclusion  already 
derived  from  (287). 


(313) 


THE    TIIKORY    AND    PRACTICE   OF   INTERPOLATION. 


IC) 


N^ow,  as  was  ])iwiously  remarked,  the  value  of  the  tirst  integral 
corresponding  to  the  lower  limit  is  usually  known  in  practical  appli- 
cations. We  shall  tlu'refore  denote  by  11^  the  value  of  f  F {T)  <IT 
which   results  when  f  is  substituted  for   T.     Then,  by   (201),  we    have 

//„   =       CF{T)dT         =J  CF(t.  +  no>)dn 

=  «,('i'\-i7^o+^.y'„+yj;'+8j;"+£jr+  .  .  .  .) 
or,  upon  restoring  the  numerical  values  of     /3,  y,  S,  .  .  .  .     from  (222), 
and  transposing, 

J^i    — h  n- -f'o  +  TJ-' o~  5-1  -^0  +  T5(r -^0  Tito  ^^0    +c;o4B(J'^o         ....  K''^*) 

(0 

which  determines  'F^,  and  hence,  also,  the  double  integral    Y,  provided 
H^  is  known.     In  practice  the  value  of  //(,  is  frequently  zero. 

Using  (314)  in  conjunction  with  the  relations  (313),  we  obtain 
the  several  groups  of  quadrature  formulae  given  below  : 

// 

-t^l     — r  T-'^o  I     T5  -^  0 — 54  -^0    T  755  -"o  Till)  ^0    T  1I04  sTJ  ^0  .... 

(1) 

rr^x'T)rf2'-   =  u>^C  j'F{t  +  uw)<hr 
I  J.^      0      I      I     1,^1^      I         i'  1       ,/"i        10       4'"  :*        //'Vj_        8G3        y/V 

''1    = \-  ^  ^0+  "12  ^  0~   *I+  ^u  "T"  T2U  -^0    — T^a  ^0  ^  ff(y4S(T'^0—    .      •      •     . 

a» 


(315) 


(316) 


-  AiU  i-Jir-^'J)  +  ^},U  i^l-^,)  - 


H. 


-tl      = b  ^  -1^0+    1-2    ^  0  ^i    ^«    +    T2T)    ^0     —  TSU  ^0  +   !ru48T)    ^0  •       •       ' 

rrix'r)(Zr-  =  o>- ('(>(<;+ ».a,)(7)r 

=  or\c'F„,-"F„^^+^,(F,-F,:)-^i^(j['-j::)+,u(/ii"-^n 
-  ^§fk  (4^ -^!n  + iritis  (^r -^i)  -  •  ■  •  -I 


(317) 


-H'„ 


^'1     = h  t  -'^0+  TI  -^  0         24  -^0+  -7'2<5  '^0  ro  iJ  ^0  +  ff  0  4S0  ^0 

(    )i^(2')(Zr-   =   coH       /^ (/!  +  ««,)  rf»r 


(318) 


=    «>^K"^-."+l-"^..'+l)+  TV(i^,..-^',.)  -^U(^n"-^'^')+  j4ttK"-^;.'') 


168  THE    THEORY   AND    PRACTICE    OF   INTERPOLATION. 

The  foregoing  formulae  arc  applicable  when  the  upper  limit  falls 
near  the  heginnhxi  of  the  tabular  serit's.  When  the  u|)per  iinut.s  falls 
at  or  neai-  the  (nd  of  the  given  series,  the  ibllowing  formulae  —  like- 
wise derived  from   (313) — may  be  employed  : 

w 

C("F"(T')dT-   =   o^-'CJ'F(t+nw),bi'  \       (319) 

jjF{T)JT-  =   wj   I /•'(/+)( a,) </«-  \       (320) 

TT  ^ 

/ c  0  j_  1  ?("' _i_   1     /'        1     /"  X   1 'I   //'" ^    //'"-I-    '<fia-_/-/'_ 

'-T.      = h  ^-'^0+    12  -'  0~   21  ^0    +    750   ■^O  TCIJ-'O   +    (10  4  50   ■^o  .... 

(1) 

rr^'(V)(ZT'^  =  o)- (  p''((:  +  «<^)(^/i.''  \       (321) 

=  co'^s  ("^,+1-  "/';,+i)  +  -i'2  (^,-  ^':.)  -  .4o-  (4-2-^;/)  -  ^i^  (^;-3+^;.") 
-  ^uhi^t,-^":)  -  ^hui/^u+^-o-  ....  I 

a"F"{T)(lT-  =  u>-f  (F(t  +  »M>)d>r-  \       (322) 

In  applications  of  all  the  preceding  formulae,  the  value  of  "1^\ 
(or  of  "Fg  when  employed)  is  wholly  ai-biti-ary,  and  therefore  may  be 
assigned  at  pleasure  in  every  case.  Hut  when  (315),  (316),  (319) 
and  (320)  are  applicable,  it  is  frequently  convenient  to  determine  "-F, 
such  that 

/(/,'  I      A'    J_        1        ,/"  1        /)'"  _l.       22  1        //iv  19       //»4_  =0 

The  formulae  in  question  then  take  the  form  as  follows  : 


THE    THEORY    AND    PUACTIOE    OF    FNTEUI'OI.ATIOX.  1()9 


jr 

-'I  —      ^       1     2   '  0  T^   1  i  '^  0        2  4  -^O    T    7  50  -^0  TOO   ^0  +   (iiJ480  • '"  •      •      ■      ■ 

II  V    —     I      A'  X       1         /"  -  I        7'"    J-      '."-'1        /I"  1!)         /'J- 

1  1  •-'  ^  (I  T^  54"  ^0  -.Mu  "o     'r  (;n4gs  -^o  (S04S  ■'-'u  T     •     •     •     • 

/>    /'r+.(u  /•  /■' 

/''(7')'/r-   =   or  /'(/ +  //<„)(/«■- 

If     —     ?4.    1    //  4_     1       /'   1       /"X     'D     /I'"  ••!        /'*■  X      s«3        /' 

-'    I   ~       ^   T  7  '  OT   1  3"  -^  0         24  -^0   T  7  25  ^'u      ~  1  B  (f  -^0   +  iS  (i  4  S3  •  ^u  ~"    •      •      •      • 

-■  1  T-.'    '  0    1^  54  0  •  'o  24  0   ■  'o     +  (SSjiSTJ  ^0    —  604  5  "^0+      •      •      •      • 

0*f4-'itu  /»    /•« 

F{T)dT-   =   (0=1  j /''(<+««)  c/«'^ 

=  <"'("^,+i+A^-45^;.'+5iuX'-BsUo^;:+ais^^:;- •  •  •  •) 

-•  1    —  T  ^  ■<  oT^    15^0        24   '-'"   T^   720  -'o  1  ilO  -^0  T  j;04«(T  -^'a  —     .... 

(O 

'//F"     I       P    X        1        //"    1        //'"X        2 '2  I        //iv  19       ,-/v    1 

-'  1    —  12^0    I     54(7 '^(i  24  (T  "^0     Tiirf4sTJ^u  ITOJB'^i)"'"     .... 

I    li^(T)f/r'-   =    <o=j    \F(t  +  nw)dn-' 

iJll'F       X       1     7*^ 1       /•/"     1       -/'"    22  1       //iv     10        ,/v  N 

'"V     -*   1+1+    15-'  i         540-^i-2         540-'i-3         5(j4»B-^i-4 11045-^1-5 ■       .       .       .    ) 

IT 

///    _     „0x4/''X     1      /' 1    -/"X     l«     J'"—    ■■'        /'"X      «o''       /» 

-'  1   —  T^  ^-'  0  T^  T2  -*  0        24  -"O    T   730  ^-"o  TiIiT    -"ll    T^  (T045?)  ^0  ~    .... 

(O 

/'A'     I      P    X        1        ,/''   1        //'"  X        22  1  /iv  10        //"X 

■'1  —     —   15-'  0  T   240  -^0  J40  ^'u     T  B04H(y  ^0    —  iTTJlS  ^u  +     .... 

i'Xr)  (IT-  =  O.M    /'(*+ "'")  f'"' 

—  i.fiC'F     X    1   A' 1     /"    1      /'"  221    ,/iv    11)     ,iv     \ 

—  <"  1^    -■  n+lT^    12-^11        240'-'n-2        240-'>i-3       C0450^ii— 1       Btr4S"'n-5 —    •      •      •    ) 


(.-^23) 


(324) 


(325) 


(326) 


The  diffeixniees  which  appear  in  tlie  Ibregoing  fornmhie,  together 
with  the  auxiliary  functions  'F  and  "F,  are  to  be  taken  according  to 
the  schedule  on  page  IGl.  The  symbol  i  denotes  a  positive  integer, 
while  'II  designates  a  fractional  or  mixed  number  :  so  that  all  functions 
and  differences  whose  subscripts  involve  n  must  be  derived  from 
their  respective  series  by  inter j)olation.  Finally,  the  quantity  //„  de- 
notes—  as  previously  defined  —  the  value  of  j  F (T)dT  when  t  is 
substituted  for   T  :     so  that  we  have 


JL  = 


fF^T)dl^l  (327) 

'  Jt=i 


It  may  happen  occasionally  that  the  value  of  JT^  is  luiknown, 
while  the  value  of  {F{T)dT  corresponding  to  T  = /-(-mw  is 
known    for    a   particular    value    of    n.     Denoting   this    quantity  by  H^, 


170 


THE    THEORY   AND    PKACTICE    OF   INTKIM'OLATION. 


we   may,  by  any  one   of  the  foregoing    methods,  compute   the    definite 

F{T)dT  = 


integral 


//..  -  IL 


and  hence  ihid 


(327o) 


H^  =   7/,  -  A" 


with   which  value  we  proceed  a«  before. 

Several  examples  will  now  be   solved    as    an    exercise    to  illustrate 
the  formulae  given  above. 

Example  I.  —  Let  it  be  required  to  find 

Y  =  C  i'cosTdT- 
ow  the  supposition  that    |'cosTfZJ'=2    when    T  =  0. 

AVe  tabulate  and  difference  the  following  values  of  i^(T)  =  cos  T  : 


T 

I'F 

'F 

F(T)=cosT 

J' 

J" 

J'" 

Jiv 

O 

0 
10 
20 
30 
40 
50 
60 
70 
80 
90 

0.00000 

11.95916 

24.90313 

38.78679 

53.53648 

69.05221 

85.21073 

101.86925 

118.86979 

136.04398 

11.95916 
12.94397 
13.88366 
14.74969 
15.51573 
16.15852 
16.65852 
17.00054 
17.17419 

1.00000 
0.98481 
0.93969 
0.80603 
0.76604 
0.64279 
0.50000 
0.34202 
0.17365 
0.00000 

-  1519 

4512 

7366 

9999 

12325 

14279 

15798 

16837 

-17365 

-2993 
2854 
2633 
2326 
1954 
1519 
1039 

-  528 

+  139 
221 
307 
372 
435 
480 

+  511 

+  82 
86 
65 
63 
45 

+  31 

Accordingly,  we  have 


t  =  0° 


=   10°   = 


18 


7/„ 


i  =   9 


Proceeding  by   ('^10),  the  computation  of  'F^  is  as  follows 


Ih^'-   = 

+  11.45915.6 

/;  =  +1.00000 

+  i  ^;  = 

+ 

0.50000.0 

--^'o=  - 

1519 

+  ,V  ^^'o  = 

— 

120.6 

<=  - 

2993 

-  -h  -k;  = 

+ 

124.7 

<'=  + 

139 

+  7¥ff  -^o"  = 

+ 

3.7 

J!,-=  + 

82 

.3   ,i\V    

— 

1.5 

.-.  'F,   = 

+  11.95916 

THE    THEOltY    AND    PRACTICE    OP    INTERPOLATION. 


171 


Tilt!  column  F  is  now  completed  by  8ucces^sive  adtlitioiis  ;  Jicnce, 
also,  the  colunm  "F,  having  first  assumed  "Fi  =  0.  AVheuce,  by 
(319),  the  remainder  of  the  computation  is  as  follows  : 


"h\^  =    +136.04398 

"t\  = 

o.(h;)OOo 

("F,„-"F,)   =  +136.04398 

F^   =     0.00000 

^0  = 

+  1.00000 

+  ^5  (-^9-^0)   =  -  0.08333.3 

J'^'  =    -            528 

4'  = 

-   2993 

-  .-1^  (^i'-^;')  =  -      10.3 

4'."=  +     511 

<'  = 

+    139 

-  jItt  (^;"  +  /^;')=  -       2.7 

jr=  +        .31 

^i^  = 

+    82 

-iT§liTr(4^-4'')  =  +       0.2 

log^  = 

2.1334129 

^  =  +135.96052 

log  ui'    = 

8.4837548 

log  Y   = 

0.6171677 

.-.  Y   =     4.141595 

To  verify  this  result,  we  have 


=  C  fees  Ti 
.'  Jo 


COS  TdT  =   sin  T+  C 


IT-'  = 


-con  I'+C'T 


=  l  +  iC\ 


where  C  is  the  constant  of  tlie  first  integration.     To  determine   (J,  the 
first  of  these  relations  grives 


If.  = 


sin  T+  C 


=    C 


whence 

and  therefore 


C  =  2 
r  =   1  +  ,r  =   4.141593 


Example  II.  —  Compute  the  value  of 

Y  =J£t^^^IT^ 

which  corresponds  to     11^  =  0. 

Here  we  tabulate  and  difference     F(T) 


T~^     as  below 


T 

"F 

IF 

F{T)  =  T-"' 

J' 

J" 

J'" 

2.0 
2.1 
2.2 
2.3 
2.4 
2.5 

-0.02082 

+  0.10210 

0.45178 

1.00807 

1.75340 

+  2.67234 

+  0.12292 
0.34968 
0.55629 
0.74533 

+  0.91894 

0.25000 
0.22676 
0.20661 
0.18904 
0.17361 
0.16000 

-2324 
2015 
1757 
1543 

-1361 

+  309 
258 
214 

+  182 

-51 

44 

-32 

172 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


AVc   have,  tluMefore, 

t  =  2.0 


U.l 


IL  =  0 


whence,    proceeding    by   (o26),  the  comi)utation    of    i^,    and  "F^   is    as 

follows  : 

+    I    i?;  =    +0.12500  

+  ^V  J\  =    -        19.3.7  -    ,'.,  /;   =    -0.02083.3 

-/.j;,'  =    -  12.9  +.U<  =    +  1-3 


0.2 


.-.  'F^  =    +0.12292  .-.  "F^  =    —0.02082 

From  the  completed  table  we  now  iind 


n  =  (2.468 -2.0)  H- 0.1 

=  4.68   =  o  -  0.32 

F„  =  +0.16418 

//"    =  +        191 


^::-.= 


36 


"/;+!  =  +2.36025.6 

+    J.   F„  =  +       1368.2 

—  5^5  ^»-2  =  —                    0.8 

-^U^'Z,  =  +         0.1 


2  =    +2.37393 
.-.    Y  =    +0.0237393 


This  result  is  easily  verified,  foi-  we  have 

Cl'-'dT  =   -    ,+  C 


Y  = 


-  losr,.  r+  CT 


=  -  los,  1.2.34  +  0.468  C 


also 


0   =   //„   = 


-j,+  C 


c  =  ^ 


=  -i+c 


Hence 


Y  =    -log,1.234  + 0.234   =    -0.2102609+0.234   =    +0.0237391 

with  which  the  above  result  substantially  ayrees. 

Example  III.  —  From   the   table    of  the   preceding   example,   find 
the  value  of 

Y  =  C  Cr-'dT'' 


THE    TIIKOKY    AND   TltACTICE    OF   INTERPOLATION. 


173 


Here  we  employ  formnlu  (324),  in  which  we  take 

2.15  -  2.0 


0.1 


We  therefore  obtain 


(«+l=  2+i) 

F„     =  +0.21633 

./j;;  =  +      235 

.;;;'  =  -       38 


=   1.50   =   1  + 


"/^„^j  =    +0.24992.0 

+  ^ij  F,  =    +       1802.8 

_4„j;'  =  -         1.0 

+  ^u^:'  =  -       0-2 


V  =    +0.26794 
.-.    Y  =    +0.0026794 

The  true  mathematical  value  of  Y  is  — 

Y  =  0.075  -  log,  1.075   =    +0.0026793 

78.  Double  Integration  as  Based  upon  Stirling's  and  Bessel's 
Formulae  of  Interpolation.  —  Let  the  schedule  of  functions  (including 
'F  and  "F)  and  differences  to  be  used  in  the  subsequent  formulae  of 
quadrature  be  as  follows  : 


T 

I'F 

'F 

F(T) 

J' 

J" 

J'" 

t  -2^ 

P-2 

J'     , 

^-'2 

-^"3 

t  —  u> 

"F  , 

F  , 

■^"1 

<F  , 

•J'     , 

j"[ 

t 

,ip 

F, 

■Jo' 

'F, 

J\ 

4" 

t    +   0. 

tip 

F. 

^;' 

'F. 

J'. 

ji" 

t  +  2u, 

"F„ 

F-i 

^•r 

• 

t  +  {l-l)^ 

"F,^r 

■ 

/'',-! 

■4'i     . 

^;-i 

^.'-j 

t  +  iut 

"F, 

F. 

j[' 

t  +  (7  +  1)0, 

lip 

ip 

^,+X 

^[\. 

^.';'j 

t+  (i+2)<u 

-^'+2 

4';. 

From  the  form  of  (263)  it  follows  that  the  expression  for  the  in- 
definite integral  of    F{t-\-nw)dn     is  — 


CF{t+7i<u)dti  =  e{») 


(328) 


174 


THE    TIIEOKY    AND   PRACTICE    OF   INTERPOLATION. 


Now,  by   (2()()),  we  have 
and  hence  the  pi-eceding  equation  becomes 
For  brevity,  let  us  put 

«    =   +  ii  i    =    -  ^W^  '•   =   +  jrn'VVaU 

and   (328(^/)   ma}-  be  written 

^F{t.+  nio)dn   =  f-F„dn   =    'F„  +  a  /'„  +  iJ,',"  +  cJl  + 


(328a) 


(329) 


(330) 


the   constant   of  integration   being  contained  in  'F„.     Multiplying  this 
equation  by  dii,  and  integrating,  we  get 


f  CF(t+  n (o)  tin-   =  C'F„ d„  +  a  il '„  dn  +  h  i^  .  J',; 'dn  +  c  Cjl dn  + 


(331) 


Applying    formula    (330)    successively    to    each    of    the    integrals 
expressed  in  the  second  member  of  (331),  we  obtain 


C  CF{t  +  nw)dn"  =  "F„  +  aF„  +  hJ',;  +  czl]:  +  . 

+  a(F„+a/Ji:  +  bJ':+  . 

+  b(J',;+aJ]:+  . 

+c(j\:+  . 


+ 


=   "F„+2aF„+(a-+2/))J:;+2(al>  +  c)J'':;+.    .    .    . 

Whence,  restoring  the  values  of     a,  h,  c,  .  .  .  .     from  (329),  and 
reducing,  we  obtain 

JJ^(i  +  «a.)fZ«^  =   "i^„+ .'.i?',-,'^ //;.'+ ^^\V^  Jj.'-  .    .    .    .  (332) 

If,  as  in  (327),  we  denote  by  //„  the  value  of  fF(T)  (IT    which 
obtains  for     T  =  t,     then,  by  (328),  we  have 


//„ 


'^   =   \' CF(T}dT  =  J    CF{t  +  nM,)dn 

and  hence,  by   (272), 


0.0  (0) 


(333) 


THE    TIIKOIIY    AND   ritAOTICE    OF    INTERPOLATION.  175 

Upon  substituting-     i  =  0     in  the  first  of  equations   (269),  we  get 

i'F,)   =  i{'F_,+  'F,)   =   'F,-iF^ 
whicli,  together  with  (333),  gives 

'F,  =   — °  +  i  K  +  iV  (-J'o)  -  tV-o  (4',")  +  r^Uh  i^l)  -  .    .    .    .  (334) 

where    the  differences   enclosed   within    parentheses   are   means   of  the 
corresponding-  tabular  quantities,  as  defined  by   (269). 

By  employing  simultaneously  the  relations  (332)  and  (334),  and 
assigning  various  limits  to  the  integral,  we  obtain  the  following  group 
of  formulae  : 

'F,  =  —'+i  F^+  tV  (J'o)  -  7'5'u  (4;") + a^  h  i^l)  -  .  .  .  .  \ 

(O  I 

C  ("F{T)dT-  =   0,-C  CF{f  +  no,)  dn-  }        ^^^^^ 

=  a>^K"^-"^)+i\(^-^)-^in(4'-^:')+^AViy(4"-^ir)-  .    .    -I         / 

'F     —        0  X  4  A'  J-    1    (  1' \  —    11    {.i"'\±.     1  fn      { /P\  —  \ 

CCF{T)dT-    =    ,j'ffi\l  +  Hu.)dn'  )         (^^^) 

=  co'i("i^,-"i^o)4-iV(^'^..-^';)--io(4';-4;')+irnV^5(4:-40- •  •  -i     / 

'F,  =  ^+iF„+  rV  (^'0)  -  tVo  (^o")  +  ^h'ih  (^0)  -  .    .    .    .  \ 

<"   '  f 

r  rF("r)  rf r^  =  <o-f  Cf (t  +  no,)  dn^-  I       (^^'') 

=  a>1("^,-"^'„)+TV(^<-^,.)-.iTT(4'-4'.')+5^VHw(4'-^L')-    •      •      -I  / 

'F^  =  ^  +  i^';+TV(-''o)-TViT(4")  +  ai^n(4o-  •  •  •  •  \ 

f  r^( T)  ,Z r=  =  .0=  f  CF(t  +  nu)  d7i^  I        ^^^^^ 

=  a,^K"^."-"^,')+i'5(^;"-^.)-5i5(^;.''-^:o+ir^Vo(-^i^'-4;)-.  .  i     / 

In  the  jjreceding  group  the  value  of  '' F^  is  wholly  arbitrary.  We 
may,  however,  determine  the  quantity  "F^  such  that  the  sum  of  the 
terms  in  (335)  and  (336)  having  the  subscript  zero  will  vanish  :  these 
formulae  may  therefore  be  written  — 


176 


THE    THEORY    AND    PRACTICE    OF   INTERPOLATION. 


-   '  ta 

F{T)dT-  =   «,-j   I  F(i;  +  wco)(/w--^ 

(U 

"P       —       _     1       P   J-        1         7"    _  ■■'  I         y/'*   -I- 


(339) 


(340) 


Let  us  now  denote  the  second  member  of  (332)  by  y  {11)  ;  that 
is,  lot  us  put 

7(m)    =    "^,.+  1^5^,  -sinX+rrriVss'^;:-  ■    •    ■    •  (341) 

Making-     n:=l-\-l,     this  becomes 

yii+i)   =   "F,+i  +  i\  ^+s  -  5^  ■^;;j  +  TTiJVss  ^^i  -  .    .    .    .  (342) 

It  will  be  observed  from  the  foregoing  schedule  that  "i'Vi?  ^'+*' 
j",.,.^,  ....  are  not  explicitly  given,  but  must  be  derived  from  their 
respective  series  by  interpolation  to  haJres.  For  tliis  purpose,  let  us 
put,  in  analogy  with   (269), 


(i^,+5)    =  i{F^+F.+.) 


(4-;*)  =  i(^,''+4';o 


(343) 


then,  after  the  manner  of  (270),  we  shall  have 

"i^,.+,  =  {"F.^0-  H-p.+i)  +  tIsC^'Is)  -  TT/^^i  (4;«)  + 
^+»  =  (^+j  -  h  (•^;;s)  +  tI^  (-^i+j)  - 

(j;+j)  -     i  (^;;o  + 


4;* 


(344) 


Upon    sul)stituting   these    expressions    in    the    second    member    of 
(.342),  and  reducing,  we  find 


y  (i  +  i)  =  i"F,^0  -  ^\  (F,+i)  +  tHit  (^Hi)  -  T/AVtf  (4+j)  + 


(345) 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATrON. 


177 


Again,  by  means  of  (.332)  and  (341),  we  derive 

C  CF(t  +  nw)(/n-  =   y(n")  ~y(7i') 


(346) 


Finally,    denoting    by    //_+     the    value     of      fF(T)dT      when 
T=t—},cj,     we  shall  have,  by  (328^0' 


jr_ 


<f' 


(T)dT 


=     J     CF(t+7lu>) 


(In 


„,(IP        X      I      ,7'         '7        /I'"  J.         .■)  G  7  //v       \ 


which  gives 


'P  —  <  1/1'         J_         17       A'"—  ■■"5  7  y^V       4_ 


(347) 


By  assigning  various  values  to  the  limits  n  and  n'  in  (346),  and 
employing  either  (341)  or  (345)  as  required  in  each  particular  case  ; 
and  finally,  by  using  either  (334)  or  (347)  to  determine  the  series 
'F,  according  as  the  assigned  lower  limit  is  not  or  is  equal  to 
we  derive  the  group  of  formulae  given  below  : 


1 

2  > 


II V  1    jf  s.     1     /I" 3  1      A"  ±. 

/»  /•H-(*+i)(u  /•  /•i+J 

j   JF(T}dT''  =  «.M  ji^(;!  +  woj)(ZM^ 

'^!    =     ^"+i^„+TV(^'o)-T'5V(X')+aik(4)--      •      •      • 

"F^   =   ««y  convenient  value ;    arbitrarily  assigned. 

(  ji^(r)(Zr2  =  wM  ( /''(!;+ WO))  <^ra^ 

=  -^K"^+5)-5'4(^+j  +  tU«  (-/:;«) -T^'5^(^i;o+  •  •  • 

II  V  1      P    4.       I       //" 3  1         //iv  . 

ITf     —    zi  I     //'       -4-       17       A'"   3  (i  7  //'J. 

rr^x"r)(/T-  =  <o=rr>(!!+wco)r;«-^  . 

—  ,.?i"F  X-    1     T*" 1     //"  X      3  1      //iv  \ 

—  <"  (,    -'  !  +   IJ  -'^i       540  ^i     +  STJ4STr  ^i     —    ■      •      •      •    j 


(348) 


(349) 


(350) 


178 


THE    TllEOKY   AND    rKACTICE    OP   INTEEPOLATION. 


H 


IW       -i  1        ,/'         I  17         /*'"    3(17  /f     _1_ 

w 


(351) 


'F_ 


H 


-i 


(D 


54  '^  -1  +  57ffir  ^— J 


■■!  (;  7 
J  —  TJtTTCSS 


^1>+ 


"ifl   =   any  convenient  value;    arhitrarily  assirjtied. 


The  last  formula  may  also  bo  written  in  the  following  form 


(352) 


/f_. 


3  G  7         A''     X- 


F{T)dT-  =    u)M   j  F(C  +  »<o)(/«- 


(353) 


I 


It  may  be  well  to  again  point  out  the  fact  that  the  functions  and 
differences  enclosed  within  parentheses  denote  the  means  of  coi-re- 
sponding  tabular  quantities,  as  defined  by  (269)  and  (343).  Further, 
that  //(,  and  i/lj  denote  the  values  of  the  first  integral  of  F{T)  when 
for  T  we  substitute  t  and  t — ^o),  respectively.  Finally,  we  may  add 
that  if  in  any  case  Hp  is  given  and  H^  required,  it  is  only  necessary 
to  compute 

F(T)dT  =   n^-IT, 

and  thence  find  /       (^^^) 

If,  =  H^-X 

In  the  process  of  double  integration  by  mechanical  quadrature  it 
is  sometimes  convenient  to  tabulate,  not  the  given  function,  but  w" 
times  that  quantity.  By  this  means  all  differences  are  multiplied  by 
w^,  and  thus  the  final  multiplication  by  that  factor  is  avoided.  How- 
ever, in  order  that  the  quantities  'F  and  "F  shall  be  multiplied  by  the 

77" 

same  factorj  it  is  evident  that  tlie  independent  term    —    (which    has    the 


THE    THEORY   AND    PRACTICE    OF    INTERPOLATION. 


170 


same  fixed  value  wlietlier  we  tabulate  F {T)  or  or F {!''))  must  like- 
wise be  multiplied  by  w~ :  so  that,  [)roeeeding  by  this  method,  it 
becomes  necessary  to  take  w//  in   place  of  the  term     -    which  occurs 

-'  ^  CO 

in  all  the  preceding  formulae.  The  computer  is  cautioned  against 
neglecting  this  precept  in  case  he  tabulates  oi^F^T)  instead  of  the 
given  function    F(T). 

We  close  the  chapter  with   several   examples  which    illustrate   the 
formulae  ffiven  above. 


Example  I.  —  Find  the  value  of 


Y  = 


2.6 

2TdT- 


on  the  supposition  that  the  first  integral  vanishes  for     T 
We  tabulate  the  given  function  as  below  : 


9  9 


T 

iiF 

'F 

—2T 

J' 

A" 

J'" 

Jiv 

2.0 
2.1 
2.2 
2.3 
2.4 
2.5 
2.6 
2.7 
2.8 

0.000000 

-0.063375 

0.243017 

0.527697 

-0.907502 

-0.063375 
0.179642 
0.284680 

-0.379805 

-0.160000 
0.143501 
0.129011 
0.116267 
0.105038 
0.095125 
0.086353 
0.078575 

-0.071661 

+  16499 

14490 

12744 

11229 

9913 

8772 

7778 

+  6914 

-2009 
1746 
1515 
1316 
1141 
994 

-  864 

+  263 
231 
199 
175 
147 

+ 130 

-32 
32 
24 
28 

-17 

Here  we  have 

t  =  2.2  o)  =  0.1 

whence,  employing  (335),  we  find 


i  =  4 


H,  =  0 


F^  =    -0.129011 
(J'„)   =    +      13617 


K")  = 


247 


+  i  F„  =  -0.064505.5 
+  tV  (-''o)  =  +  1134.7 
-j\M^'o")  =    -  3.8 


.-.  'F,  = 


■  0.063375 


Assuming  "F^^O,  we  complete  the  table  as  shown  above  ;   thence, 
proceeding  by  (335),  we  obtain 


DR.  GEORGE  F.  McEWtN 
180  THE    THEOKY   AND   PBACTICE    OF   INTERPOLATION. 


I'F^  =    -0.907502         "F. 


Fi  =    -0.08G353 


^I' 


994 


4','  = 


0.000000 

-0.129011 

1746 


-0.907502 
+        3554.8 
-  3.1 


.:    Y 


-0.903950 
-0.00903950 


Verification  :     Integrating  directly,  we  have 
J  (1  +  T^)-       1+r-  "^ 


whence 


r  = 


0  =  //,  = 


taB -iT+CT 


(1  +  T-y^ 


+  C 


C  =   -0.17123288 


Finally,  using  the  relation 

tan"'  a  —  tau~'  h  =  tan~' 
the  preceding  expression  for   Y  becomes 

0.4 


a  —  h 


which  gives 


F  =  taii-Mg^l+0.4C 


Y  =    -0.00903949 


ExAivn'LE  11.  —  From  the  table  of  the  preceding  example,  compute 


_  rpiTdT- 


.23 

Here  we  employ  (349),  taking 

i!  =  2.2  i  =  3  //o  =  0  M  =  (2.23-2.2)-:- 0.1  =  0.30 

Thus  we  find 


("7?;^)   =   -0.717599.5 
-    i,  (^8»)  =   +        3780.8 


(F,i)   =    -0.09(t7;i9 

(j^;)  =  -      IOCS 


-0.713828.2 


Also 


whence 


THE    THEORY   AND   PRACTICE    OF   INTERPOLATION. 


181 


(«  =  0.30) 

F„  =  -0.125016 

j;;=  _    1673 

-  "K 

-  tV  K 

+  ^h^:: 

=  +0.006077.9 
=  +0.010418.0 
=  -      7.0 

Y  = 

-0.00697339 

=  +0.016488.9 
=  -0.697339 

Verifying  this  result  as  in  the  i)receding  example,  we  find 

-1  f  '^■ss  ^ 


Y  =  tan- 


6.6865y 


+  0.32C 


-0.00697338 


Example  III.  —  Let  it  be  required  to  find 


T  = 


_    /Vj/cosTf/r- 
/  /       sin^  T 

•^  •.'■mo 


assuming  that   the   first  integral  =  231  when  T  =  30°  ;    M  being  the 
modulus  of  the  common  system  of  logarithms. 

Here  we  tabulate  F(T)  =  — w^J/cos  Tcsc^T  for  T=  20°,  24°, 
28°,  ....  60°  ;  thus  avoiding  the  final  multiplication  by  w^.  Since 
6j  =  4°  ^  TT  +-  45,    we  find 

log  ,s,-M  =  7.325659  -  10 
Our  table  is  therefore  as  follows  : 


T 

np 

'F 

F(T)  = 
—  cj-M  cos  T  csc-T 

J' 

J" 

J'" 

Jiv 

o 

20 
24 
28 
32 
36 
40 
44 
48 
52 
56 
60 

+  0.029974 
0.084135 
0.133339 
0.178619 
0.220744 

+  0.260304 

+  0.060553 
0.054161 
0.049204 
0.045280 
0.042125 

+  0.039560 

-0.017004 
0.011689 
0.008480 
0.006392 
0.004957 
0.003924 
0.003155 
0.002565 
0.002099 
0.001722 

-0.001411 

+  5315 

3209 

2088 

1435 

1033 

769 

590 

466 

377 

+  311 

-  2106 
1121 

653 
402 
264 
179 
124 
89 

-  66 

+  985 

468 

251 

138 

85 

55 

35 

+  23 

-517 

217 

113 

53 

30 

20 

-  12 

182  THE    THEOKY   AND   PEAOTICE    OF   INTERPOLATION. 

We  proceed  by  formula  (353),  taking  as  oui-  data 

f  =  32°  «.  =      4°  =  ,r-:-45 

i  =     i  H-i  =  2.1/  =  0.8G8589 

Whence,  observing  that  we  must  now  take  coll-i  instead    of  tlie   tciiii 
H-i-^w  in   (353),  the  computation  of 'i^'-j  is  as  follows: 

log  a.//_s  =  8.782752  a,//_,   =    +0.060639.0 

//ij  =      +2088  -.^-lU  =    -  87.0 

z?-   =      +    468  +.}a'J-i  =    +  1-4 


And  for  "F„  we  find 


(F-d 

= 

-0.00 

7436 

{^J-d 

= 

— 

887 

i^J'-d 

= 

— 

367 

•.  'F_i  =    +0.060553.4 


i  'F_,    =    +0.030276.7 
+  ,ij  (F_0   =    -  309.8 


.-.  "i>;  =    +0.029974 

Upon    completing    the    table    as    shown    above,  and  continuing  the 
computation  by  (353),  we  obtain 

(i   =   4)  ("iJ^j)   =  +0.240524.0 

(F,,)   =    -0.002332  -j',(^40   =  +  97.2 

(j;;)   =    -  106  +^i|^(z/:0   =  -  0-9 

.-.    Y  =  +0.240620 

We  easily  verify  this  result  analytically  as  follows  : 

-M  cos  TclT  M 


J  sin=  T  sin  2' 

=  .1/iog,  tan  i  r  +  cr  +  C 


// 


sin^  T  sin  2' 

■  3/cosraT- 

""su?r 

=  log,„taiii2'+  CT+  C 


:    Y 


logi„tani  T  +  CT 


r=BO«=T^'gT 


But 


.-.   C  =  0 

.-.    F=  logi„tanr^j  -logiotanT— j 


THE    TIIEOKY   AND   PKACTICK    OF    INTERPOLATION.  183 

Now  we  find 

log  tan  25°  =  9.068072.5  -  10 
los  tan  15°   =   9.428052.5  -  10 


.-.    Y  =  0.240620 
which  agrees  exactly  with  the  former  result. 

Example  IV.  —  From  the  table  and  data  of  Examjile  III,  compute 
the  integral 


_  /Vj/cosra: 

//      sm'T 


Here  we  employ   (351),  taking  i  =i  32°  as  before  ;    we  then  have 
for  the  value  of  n  at  the  uppei-  limit, 

n  =   (45° -32°)-:- 4°  ==  3.25   =  3  +  0.25 

We  therefore  obtain 

"F^  =  +0.189420.3 

7*;   =    -0.002993  +,V-f»  =  -  249.4 

A':^  =    -  163  -gj-ff^'.'   =  +  0-7 

.-.    r  =  +0.189172 

Verifying  this  result  as  in  the  last  example,  we  find 

Y  =  Iogi„tan22°30'-logiotanl5°  =    +0.189172 

Example  V.  —  As  a  final  exercise,  combining  both  single  and 
double  integration,  and  illustrating,  moreover,  the  use  of  formula  (339) 
when  several  values  are  assigned  in  succession  to  the  integer  ?',  we 
shall  conclude  these  examples  with  a  complete  and  detailed  solution 
of  the  following  problem  : 

A  particle  P  of  unit  mass  is  impelled  along  a  straight  line  AB 
by  a  varying  force  whose  expression  is  20000  T ~^ ;  where  T  is  the 
time  in  seconds  after  a  definite  epoch,  and  the  implied  unit  of  length 
is  one  foot.  It  is  required  to  find  by  quadratures  the  velocity,  v^  and 
the  distance,   AP  =  a?,   for  the  times 

T  =   102,  104,  106,  108  and  110  seconds,  respectively; 

assuming  that  v^  =  O.G  feet  per  second  and  x^  =:  8  feet  when  T^  =  100 
seconds. 


184 


THE  THEORY  AND  PRACTICE  OE  INTERPOLATION. 


Since  the  mass  of  P  is  unity,  we  have,  simply, 

d-x    _    20000 

whence  by  a  single  integration 

fZ.-r  /20000(Zr 

+  "o 


(«) 


and  by  double  integration 


rfiomodT" 


w 


We  shall  iii'st  compute  the  required  values  of  x  as  given  by  equa- 
tion (^),  effecting  the  double  integration  by  means  of  (339).  The 
details  of  the  computation  are  shown  in  the  following  table  : 

Table  (A). 


r 


96 
98 
100 
102 
104 
106 
108 
110 
112 
114 


40000  T-" 


0.04521 
.04250 
.04000 
.03769 
.03556 
.03358 
.03175 
.03005 
.02847 

0.02700 


-271 
250 
231 
213 
198 
183 
170 
158 

-147 


+  21 
19 
IS 
15 
15 
13 
12 

+  11 


+  0.53730 
.57980 
.61980 
.65749 
.69305 
.72663 
.75838 
.78843 

+  0.81690 


''F  +  ix,=  a 


+  3.99667 
4.61647 
5.27396 
5.96701 
6.69364 

+  7.46202 


+  I'a  F=  h 


+  0.00333 
314 
296 

280 

265 

+  0.00250 


x  =  a-\-h 


4.00000 
4.61961 
5.27692 
5.96981 
6.69629 
7.45452 


8.0000 
9.2392 
10.5538 
11.9396 
13.3926 
14.9090 


Since  we  shall  afterwards  use  this  same  table  in  finding  v  by 
single  integration,  it  is  here  convenient  to  tabulate  &>  times  the  given 
function  :  thus  avoiding  the  final  multiplication  by  w  in  computing  v, 
and  reducing  the  corresponding  factor  in  the  case  of  x  from  w^  to  w. 
Accordingly,  we  tabulate  under  F {T)  the  function 

F(T)  E  200000)2'-^  =  40000 r-= 

Assume    t  =  100,    aud   proceed   by  (339).     To  determine  'i^^,   it 

must   be    observed   that   since     F{T),  j',  j",  ....      already  contain 

H 

the  factor  w,  it  is  here  necessary  to  multiply  the  independent  term  — ^ 


THE    THEORY   AND   PRACTICE   OF    USTTERPOLATION.  185 

TT 

by  the  same  factor:  so  that,  writinj?  i\{=  IT^  for  "  in  the  first 
equation  of  (339),  and  omitting  insensible  terms,  we  have 

%   =  V,  +  kF,^^\{J\)  (y) 

Hence,  substituting  r„  =  0.(),  F^  =  0.04000,  (j'„)  =  .]  (  r  ,-|- j;)  = 
—0.00240,  we  find  'F),  =  +0.G1980,  and  thus  complete  the  series 
'F  as  given  above. 

The  second  equation  of  (339)  gives  simply,  "i^„  =^  — J.j  i^„,  the 
term  in  ./'  being  insensible.  But  since,  by  equation  (/3),  we  should 
afterwards  have  to  add  the  constant  ,r„  to  each  computed  value  of  the 
double  integral  taken  from  T^  to  T,  it  is  expedient  to  tabulate  in 
place  of  "F^^  the  quantity 

"F.-\--  =   "F^+h-'-o  =    --hF,+  ^.^  =  4.0  -0.00333   =  +3.99G67 

and  thence  complete  the  series  as  given  under  "F -\-  \  x^  rE  (t.  The 
reason  for  this  procedure  is  easily  made  aj^parent :  for  the  final  equa- 
tion of  (339)   gives  (since  w^  must  now  be  replaced  by  w) 


// 


20000(^T2 

3 =     „,("i^,+  -,Vi^,) 


and  substituting  this  expression  in  equation   (yS),  we  obtain 


y. 


X   =   0,  ("F,  +  -r'i  F,)  +  .r„   =   „,  ("F,  +  -°  +  -rV  F,)  (8) 


O) 


Therefore,  upon  forming  the  column     -j-iV-^^=  ^j     ^^  given  above, 
we  have  from  (S) 

ix   =   "F,  +  i.r„+^y^F,   =   a  +  h 

whence  the  required  values  of  x  are  derived  and  tabulated  in  the  final 
column  of  Table  (A). 


186 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


For  the  computation  of  tlie  velocity   v  we    employ  formula   (282), 
the  first  equation  of  which  gives 

or,  by  adding  i^o  to  both  members. 

But  we  shall  avoid  subsequent  additions  of  the  constant  V(„  required 
by  equation  (a),  if  we  increase  this  value  of  'F'i  by  the  term  v^  =■  0-^  j 
that  is,  if  we  take 

which  is  the  same  as  the  expression  (y),  iised  for  determining  the 
series  'F  in  Table  (A).  The  latter  series  is  therefore  to  be  employed 
in  finding  v,  the  computation  of  which  is  as  follows  ; 

Table  (B). 


T 

CF) 

(J') 

-A.(-J') 

i'={'F)-^^(J<) 

96 

+  0.51470 

+  24 

+  0.51494 

98 

.55855 

-260 

22 

.55877 

100 

.59980 

240 

20 

.00000 

102 

.63805 

222 

18 

.03883 

104 

.67527 

205 

17 

.07544 

100 

.70984 

190 

10 

.71000 

108 

.74251 

170 

15 

.74200 

110 

.77341 

104 

14 

.77355 

112 

.80207 

-152 

13 

.80280 

114 

+  0.83040 

•  ■ 

+  12 

+  0.83052 

Recalling  the  fact  that  functions  and  difterences  in  parentheses 
are  means  taken  according  to  (269),  the  method  of  forming  the 
second,  third  and  foui-th  columns  of  this  table  from  the  quantities  of 
Table  (A)  is  obvious.  Now,  since  the  fiictor  w  has  been  previously 
introduced,  the  second  equation  of  (282)   gives 

V  =   ('7^,)--,V(^'.) 

from  which  expression  the  required  values  of  v  are  computed  and  tabu- 
lated in  the  final  column  of  Table  (B). 


THE  THEORY  AND  PKACTICE  OE  INTERPOLATION. 


187 


This  completes  tlie  solution  of  the  problem.     An  interesting  check 
is  derived,  however,  by  observing-  that  equation  (a)  gives 


dT+,- 


W 


whence  x  may  be  obtained  from  the  series  v  by  single  integration. 
For  this  purpose  we  make  f{T)  =  oiV  ^=.  2v,  and  thus  form  the 
table  below  : 

Tai!le  (C). 


T 

f(T)  =  2v 

8' 

8" 

'/+■<■» 

(:f)+Xo=c 

(8') 

-TV(8')='i 

x  =  c  +  d 

96 
98 
100 
102 
104 
106 
108 
110 
112 
114 

1.0299 
1.1175 
1.2000 
1.2777 
1.3509 
1.4200 
1.4853 
1.5471 
1.6056 
1.6610 

+  876 
825 
111 
732 
691 
(;53 
618 
585 

+  554 

-51 
48 
45 
41 
38 
35 
33 

-31 

+  7.4067 

8.6067 

9.8844 

11.2353 

12.6553 

14.1406 

+  15.6877 

8.0067 
9.2455 
10.5598 
11.9453 
13.3979 
14.9141 

+  801 
754 
711 
672 
636 

+  602 

-67 
63 
59 
66 
53 

-50 

8.0000 
9.2392 
10.5539 
11.9397 
13.3926 
14.9091 

Here  again  we  take    t  =^  100,    and  employ   (282),  which  gives 

'/_,  =    -i/o+iVCS'o)   =    -0.6000  +  0.0067   =    -0.5933 

Increasing  this  value  by  Xo  =  8.0,  to  pi-ovide  for  the  constant  .r,,  in 
equation  (e),  we  get  -|-7.4067,  which  number  is  written  under  [/'-}-. ''o, 
on  the  line  t — i  w.  Completing  this  column  by  successive  additions 
of  the  functions  /,  we  next  form  the  series  of  mean  values  tabulated 
under  ('/')-(- cCq  =  c.  The  columns  (8')  and  — tV  (^')  =^  ^^''^  then 
computed,  and  finally  the  column  .c  =  c  -(-  d.  These  values  of  x 
agree  substantially  with  those  given  in  Table  (A). 

From  the  given  analytical  expression  for  the  force,  together  with 
the  initial  conditions  of  the  problem,  we  easily  find 

V   =    1.6-lOOOOr--         ,         .r   =   1.6r+  10000 r-i  -  252 

whence,  making     T  =  110,    we  obtain 

V  =  0.77355     and     x  =   14.9091 
which  further  verify  the  results  derived  by  quadratures. 


188        THE  THEORY  AND  TRACTICE  OF  INTERPOLATION. 

79.     It  is  Avorth  while  to  inquire  what  change   takes  jilace  in  the 
vahie  of  the  double  integral 


■^o' 


=  0/^^) 


dT' 


when,  in  a  particular  problem,  the  quantity  II  is  changed  from  an 
assigned  value  W  to  a  new  value  II".  This  is  easily  answered.  For, 
if  we  change  //'  to  //",  the  value  of  the  first  integral  —  corresponding 
to  any  particular  value  of  T — is  thereby  increased  by  the  quantity 
//" — //';  or,  what  amounts  to  the  same  thing,  the  constant  of  the 
first  integration,  M  in  (286a),  is  thus  increased  by  //" — //'.  There- 
fore, by  (287),  it  is  evident  that  1"  is  increased  by  the  quantity 
{ir—H)  (T"—T'). 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION.         189 


EXAMPLES. 

1.  Given  the  semi-major  axis  of  an  ellipse,  a  =  1,  and  the 
semi-minor  axis,    b  =  0.8,    to  find  the  length  of  the  elliptic  quadrant. 

A7IS.     l.J:1808. 

[Note  :  —  Take  tlie  eccentric  angle  E  as  independent  variable,  and  hence  find 

TT 

s   =  I  Vl  -  e^  cos^^  dE 
where  n  is  the  eccentricity,  and  s  the  required  length.] 

2.  Given  the  equation  of  a  cardioid,  ?-  =  1  -|-  cos  d  :  to  find,  by 
mechanical  quadrature,  the  length  of  that  part  of  the  curve  comprised 
between  the  initial  line  and  a  line  through  the  pole  at  right-angles  to 
the  initial  line.  Ans.  2.82843. 

3.  The  equation  of  a  curve  being  y  z=  x^  V2  —  sin  x ,  find  the 
area  included  between  the  curve,  the  axis  of  x,  and  the  two  oi-dinates, 

a?  =  f     and     .?;=fTr.  J.H.y.  0.180518. 

4.  Compute  the  value  of 

IT 


-//;. 


Vl  -  0.82  sin^r 

assuming  that  the  first  integral  vanishes  at  the  loAver  limit. 

Ans.  0.139727. 

5.     Given    a   curve   in    a   vertical   plane   whose   points   satisfy  the 
relation 

dhj    _    4x^-3 


DR.  GEORbh  I-.  ivicLWLN 
190  THE   TllEOKY   AND    PRACTICE    OF   INTERPOLATION. 

—  the  axis  of  y  being  vertical.  Find  the  difference  of  level  between 
two  points  whose  abscissae  arc  1.000  and  1.473,  respectively  ;  assum- 
ing the  direction  of  the  curve  to  be  horizontal  at  the  first  point. 

Am.  0.044228. 

6.  By  what  amount  would  the  preceding  result  be  changed  by 
supposing  the  tangent  to  the  curve  at  the  first  point  to  be  inclined 
45°  to  the  horizontal  ? 

[Note  :  —  This  question  should  be  answered  mentally.] 


CHAPTER  V. 

MISCELLANEOUS   PROBLEMS    AND    APPLICATIONS. 

80.  The  present  short  chapter  will  be  devoted  to  the  solution  of 
a  number  of  problems  and  examples  involving  certain  principles  and 
precepts  hitherto  established. 

81.  Problem  I.— To  _/zWZ  ,S' =  1''+ 2''-)-3''+ .  .  .  .  +  »■",  where 
Tc   and   r   are  integers. 

The  method  of  solution  is  best  illustrated  by  assigning  a  particu- 
lar value  to  k.     Thus,  let  it  be  required  to  find 

S  =  V-\-2'  +  o'+   .    .    .    .  +  r* 

We  tabulate  below  and  difference  the  values  of  T*  which  corre- 
spond to     T  ^  1,  2,  3,  4,  5  and  6.     Thus  we  find  : 


T 

'F 

F(T)  E  r< 

J' 

J" 

J"' 

Jiv 

Jv 

'K 

1 

'F, 

1 

15 

o 

>F„ 

IG 

65 

50 

60 

3 

81 

175 

110 

84 

24 

0 

4 

256 

369 

194 

108 

24 

5 

625 

671 

302 

6 

1296 

r-1 

'Fr.y 

(,-1)^ 

r 

IF, 

,.4 

Now,  by  Theorem  V,  the   4th  diflTerences  of  F{T)  are   constant, 
and  hence  the  5th  and   higher  difierences   all  vanish.     Whence,  if   we 


192 


THE    THEORY    AND   PRACTICE   OF   INTERPOLATION. 


considei"  the  auxiliary  series  'F — defined  as  in  Chapter  IV — we  shall 
have,  by  the  fundamental  formula  (73), 


^F,  =  '/;+,.+  !:(^(i5)+'±^)i!::z2)^^^^ 


+ 


,.(,_l)(,._2)(,-3) 


(60)  + 


r{r-l). 


(r-4) 


11 


(24) 


=   '^0  +  3(5  ('•  +  1)(2/-  +  1)  (3.^+3.-1) 
Therefore,  by  Theorem  I,  we  have 


r 
30 


S  =   'F^-  'F„  =  —  (r  +  l)(2r+l)(3?-^  +  3r-l) 


(355) 


which  is  the  required  expression  for  the  sum  of  the  fourth  powers  of 
the  first  r  integers. 

82.  Problem  II. —  Given  a  series  of  functions,  F_3,  F_o,  F_i,  F^, 
F^,  F2,  .  .  .  .  ,  and  an  assigned  intermediate  value,  F„  :  To  find 
the  corresponding  interval  n. 

First  Solution  :  The  simjjlest  method  is  to  determine  by  inspec- 
tion an  approximate  value  of  n,  and  then  find  by  direct  interpolation 
the  values  of  the  function  corresponding  to  three  or  four  closely  equi- 
distant values  of  n  that  shall  embrace  the  required  interval.  The  latter 
is  then  readily  found  by  a  simple  interpolation. 

Example.  —  From  the  following  ephemeris  find  the  time  when  the 
logarithm  of  3Iercurifs  distance  from  the  Earth  =  9.79G8280  :  that  is, 
given  F,,  =  9.7968280,  to  find  n.  The  tabular  quantities  are  here 
given  for  eveiy  second  Greenwich  mean  noon. 


Date 
1898 

Log.  Dist.  of 
5  from  ® 

J' 

J" 

J"' 

Jlv 

Jv 

May  8 
10 
12 
14 
16 
18 
20 

9.7500700 
9.7652375 
9.7768883 
9.7905482 
9.8057806 
9.8221946 
9.8394585 

+  91009 
110508 
130599 
152:524 
104140 

+  172639 

+  24839 
20091 
15725 
11816 

+  8499 

-4748 
43(;6 
3909 

-3317 

+  382 
457 

+  592 

+  75 
+  135 

We  observe  that  the  given  logarithm  falls  somewhere  between  the 
tabular   values   for  May  14  and   16,   and   soon   find   that   the   interval 


THE  THEORY  AND  PRACTICE  OF  INTERPOLATION. 


193 


(from  the  former  date)  is  somewhat  greater  than  0.4.  Hence  we  take 
F^  =  9.7905-182,  and  interpolate  —  hy  Bessel's  Formnla  —  tlie  functions 
corresponding  to     n  =  0.38,  0.41,  and  0.44.     Thus,  computing  and  dif- 


ferencing these  vahies,  we  find 


n 

F,. 

J' 

J" 

0.38 
0.41 
0.44 

9.7961736 
9.7966267 
9.7970810 

+  4531 
+  4543 

+  12 

Whence,  if  we  denote  by  u,'  the  interval  at  which  the  required 
function  lies  beyond  the  middle  function  in  this  new  series,  we  shall 
have,  by  neglecting  the  small  second  difference, 

n'  =  2013-^4543   =   0.44,     nearly. 

But  if  great  accuracy  is  required,  we  may  easily  take  account  of  the 
second  difference  by  the  method  of  the  corrected  first  difference  (§44). 
Thus,  in  the  last  table,  we  find  that  the  corrected  first  difference 
which  corresponds  to    n  =  0.44  is  4540  ;    hence  we  have 

»'   =  2013  -f  4540   =  0.4434 
.-.  n  =  0.41  +  0.4434  X  0.03  =   0.423302 

The  required  time  is,  therefore, 

T  =   May  14"  +  0.423302  X  48"   =   May  14'>  20"  19'"  6«.6 

83.  Second  Solution  of  Problem  II.  —  Given  F^,  to  find  the  value 
of  n. 

Let  ni  denote  an  approximate  value  of  n,  true  to  the  nearest  tenth 
of  a  unit,  and  put 

n   =   m  +  z  (356) 

Then  Ave  have 

F,   =   i-'^^,   =   F[f+{m+z)ui]   =   F[_(t  +  m,^)  +  z,^-] 

=   F{t-]rm^)  +  zu>F'  {t+m<s,)Jr'^  F"  (t  +  viu>)-[-    .... 

Since  we  have  supposed  z  not  to  exceed  0.05,  it  is  permissible  to 
neglect  z^,  z'^,  .  .  .  .  in  the  last  expression,  which  becomes,  there- 
fore, 

F,^  =   7^,„+.ta.i':,+  i.~Vi?';;  (357) 


19-1:  THE    THEORY    AND   PRACTICE    OF   INTERPOLATION. 

To  find  z  from    this    equation,  we  first    neglect   the    small  term  in 
z^,  and   thus    obtain    an    aj)proximate  value  which  we    shall  call  .*'.     In 

this  manner  we  find 

-f „  -  ^» 
X  =      "  ,„  "'  (358) 

This  approximate  value    of   z  will   now  suffice  for   substitution    in   the 
last  term  of  (357).     Accordingly,  we  obtain 

-("^^^  ■  (359) 


(360) 


s 

:     -X 

whence,  piitti 

ng 

y 

= 

\ 

we  have 

z 

— 

x  —  y 

and  equation 

(35G) 

becomes 

n   =   VI  +  X  —  y  (301) 

Finally,  to  express  F,„,  uFm,  and  at^J^m  iii  terms  of  the  differences 
of  the  given  series  F,  it  will  be  expedient  to  emjiloy  Stirling's 
Formula  of  interpolation,  together  Avith  the  expressions  for  F^  and 
F^l  as  developed  in  §61.  The  above  solution  may  then  be  expressed 
as  follows  : 

Determine  m  =   an  approximate  value  of  n,  true  to  tlie  nearest  tenth 

of  a  unit. 
Thence  find       F„==  F^  +  ma  +  B\  +  Co  +  DJ^  +  .    .    .    . 

Z>i  =    u>Fl,    =    a+mh^+  C'r+  />'</„+    .... 
D,=   o^'F:,:=    /,^+mr+   .... 

jr    ^^  )        (•■^62) 

A 

-        A 
y   =   ix'K 

and  w  =  m  +  x  —  // 

Here  the  differences  are  to  be  taken  according  to  the  schedule  on 
page  62  ;  the  coefficients  B,  C,  D,  .  .  .  .  being  taken  from  Table  II, 
and  C",  D,  .  .  .  .  from  Table  V.  Finally,  Table  VII  gives  the  value 
of  //  for  top  argument  K  and  side  argument  x  ;  observing  that  //  has 
the  same  sign  as  K. 


THE    THEORY   AND    PRACTICE    OF    INTERPOLATION.  195 

Example.  —  Same  as  in  ^82. 

Here  we  find     rn  =  OAO  ;     and  hence    take  from    the  g-ivcn  tahle, 
and  from  Tables  II  and  V,  the  quantities 


m   =       0.40                    a   =    +1444(51.5  

B  =    +0.080                 *o  =    +   l'''^2o  

C  =   -0.056                 c    =    -     4137.5  C  =   -0.08667 

D  =    -0.0056               fi'„=    +       457  I)'  =    -0.02267 

E  =    +0.01075             e    =    +       105  U'  =    +0.01440 

The  computation  of  F„,,  7)i  and  D..  by  (362)   is  therefore    as  fol- 


lows 


F,    =  9.7905482  

ma   =  +         57784.6  a   =  +144461.5                    

Bb^  =  +          1258.0  mb^  =  +     6290.0  b^  =    +15725 

Cc    ^  +            2.31.7  C'c  =  +       358.6  mr  =    -   1655 

i>J„=  -                 2.6  Z)'t/„  =  -         10.4                    

A'e    =    + lA.  E'e  ^  +           1.5                   

.-.  F^  =  9.7964755  .-.  D^  =  +151101  .-.  IK  =    +14070 

i?;.   =  9.7968280                                                     .-.  F,-fI,  =    +  3525 

Whence 

K  =   A+-A   =    +14070 -+151101    =    +0.0931 

X  =   (i.;_i';,)-i- z>,  =    +3525 -H 151101  =   +0.023329 

and  we  finally  obtain 

m   =  0.400000 

X  =  +0.023329 

(Table  VII)  -  y  =  -            26 

.-.  n  =  0.423303 

which  agrees  within  one  unit  with  the  foi'mer  result. 

84.     Problem  III. —  To    solve    any   numerical    equation    ivJiatever 
involving  hut  one  unhnown  quantitij. 

The  given  equation,  whether  simple  or  complex,  algebraic  or  trans- 
cendental, may  be  written  in  the  form 

F{T)   =  0 

The  problem  therefore    reduces   to   the  question  of  finding  n  when  F,^ 
is  known  and  equal  to  zero  —  which  is  the  same  as  Problem  II. 


196  THE    THEORY    AND   PRACTICE   OF    INTERPOLATION. 

Example.  —  Solve  the  transcendental  equation 

T  -  20°  sin  T  =  45° 

Avhere   T  is  expressed  in  degrees  of  arc. 
This  equation  may  be  written 

F(T)  =    T  -  20°  sin  r  -  45°   =  0 

which    by  trial  we  find    to   be    satisfied   by  a  value  of  T  not  far  from 
63°  ;    hence  we  tabulate  F (T)  for   T  =  62°,  63°,  and  64°,  as  follows  : 


T 

F{T) 

J' 

J" 

O 

62 
63 
64 

-o!g590 
+  0.1799 
+  1.0241 

+  8389 
+  8442 

+  53 

Here  we  have  given  F„  ^=  0,  to  find  n.  Whence,  employing  the 
corrected  Ji7-st  difference  (§45),  we  find 

1799 
r  =  63°  -  —^  X  1°  =   62°.7861 

8410  , 

8.5.  Problem  IV.  —  Gtiven  a  series  of  numerical  functions  em- 
bracing a  maximum  or  minimum  value  :  To  find  the  value  of  the 
argument  ivhich  correspotids  to  the  maximum  or  minimum  function. 

Find  by  inspection  the  tabular  function  which  falls  nearest  the 
required  maximiuu  or  minimum  value.  Call  this  tabular  function  F^. 
Then,  from  the  schedule 


T 

F(T) 

J' 

J" 

J'" 

Jiv 

t   —  <j> 

t 

t    +  O) 

F, 

a' 

b' 

d' 

we  have,  by  the  first  of  equations  (182), 

=   F'{t  +  n,^) 

{a-l  c  +.  .  .  .)+  «(i„_,i.,  d„  +.  .  .  .)  +  iM=(c-  ....)+  i  ""K-  ••••)  +  •■■• 


F'{T)   =   F'{t  +  nu>) 
1 


THE    THEORY   AKD    PRACTICE    OF   INTERPOLATION. 


197 


Therefore,  since    the    condition    of  niaximnni   or  minimum  requires 
that  F\T)  =0,  we  have,  by  neglecting  5th  differences, 


(„-i  c)  +  (/.„_  j>,  d^)n  +  ^en^  +  ,1  tiy  =   0 


(363) 


whicli  detei-mines  the  value  of  n,  and  hence,  also,  the  value  of  T,  at 
the  point  of  maximum  or  minimum  of  F(T).  This  equation  may  be 
readily  solved  by  successive  approximations,  by  first  neglecting  the 
terms  containing  n^  and  n',  and  afterwards  substituting  therein  the 
approximate  value  of  n  thus  found,  and  so  on  ;  or,  we  may  consider 
the  solution  of  (363)  from  the  standpoint  of  Problem  III,  —  which 
may  be  regarded  as  the  more  direct  of  the  two  methods. 

Example.  —  The  following  ephemeris  gives  the  log  radius  vector 
of  3I((rs  with  respect  to  the  Sun  (log  r) .  Find  the  time  of  iDerihelion 
passage  of  the  planet. 


Date 

1898 

Log  »• 

J' 

J" 

J'" 

Jiv 

April  6 
14 
22 
30 

May  8 
16 
24 

0.1416628 
0.1409303 
0.1404822 
0.1403232 
0.1404553 
0.1408772 
0.1415840 

-7325 
4481 

-1590 

+  1321 
4219 

+  7068 

+  2844 
2891 
2911 
2898 

+  2849 

+  47 
+  20 
-13 
-49 

-27 

33 

-36 

Here  we  are  required  to  find  the  instant  when  log  r  is  a  mini- 
mum. Since  it  is  evident  that  this  condition  occurs  only  a  few  hours 
from  April  30,  we  take  F„  =  0.1403232.  Whence,  from  the  above 
table,  we  find 


a  =  -  134.5 

a   —  (\  c  = 

-  135 

ho   =    +2911 

*o-i'A  = 

+2914 

(■  =  +   3.5 

^«  = 

+   2 

do  =    -     33 

if'o  = 

-   6 

and  therefore,  by  (363), 

-135  +  291 4?i  +  2)1^  -  6««  =  0 


or 


291471  =  135  -  2n-  +  6n' 


198  TILE   TIIEOKY   AND    PliACTICE    OF   INTERPOLATION. 

Neglecting  the  last  two  terms  of  this  equation,  we  have,  for  an 
approximate  value  of  n, 

n   =   135  -^  2914   =   0.040,     nearly  ; 

and  since  for  this  value  of  n  the  small  terms  sensibly  vanish,  we  obtain 
as  our  final  value 

71  =   135  -^  2914   =   0.04633 

The  date  of  perihelion  passag'e  is,  therefore, 

T  =   April  30'!  +  0.04633  X  S  X  24"  =   April  30"  8".895 

86.  PPtOHLEM  V.  —  Given  a  series  of  numerical  values  {F_s,  F_2, 
F'_i,  Fq,  Fi,  F2,  .  .  .  .)  of  any  function  F(T)  which  is  analytically 
unknown:  To  find  an  apjn'oximate  algebraic  exj^ression  for  F  (T)  in 
terms  of  the  variable  argument. 

Let  us  put 

T  =    T  -t  (364) 

and  Tayloe's  Theorem  gives 

F{T)   =  F(t  +  T)   =  F{t)  +  rF'  {t)+~  F"  (t)  +  '^  F'"  (t)  +  .  .  .  .  (365) 

Upon  substituting  in  (365)  the  expressions  for  F'{t),  F"{f), 
F"'(t),  .  .  .  .  ,     as  given  by   (I'J^'j),  we  obtain 

F(T)   =   F(f)  +  1  (a  -  i  c  +  jV  « )  ^  +  -V  (^0-  .'.  ''0  +  ■■■•)  ^' 


2^ 


+  r^(«-i«  +  -  •  ■  ■)r»+_^(f7o-.  •  ■  .)r*+-j^(e-.  .  .  .)r^+.  .  .  .        (366) 


which  expresses  F (T)  as  a  rational  integral  function  of  r,  with  known 
numerical  coefficients  ;  t  being  the  value  of  the  variable  argument 
counted  from  the  fixed  epoch  t,  as  defined  ])y   (364). 

Example.  —  From  Newcomb's  Astronomical  Constants  we  take 
the  following  table  of  the  mean  obliquity  of  the  ecliptic  (e)  for  evei-y 
fifth  century  : 


THE    THEORY   AJJD   PRACTICE    OF    INTERPOLATION. 


199 


Year 

Obliquity 

J' 

J" 

J'" 

0 
500 
1000 
1500 
2000 
2500 

0           1            u 

23  41  43.78 
37  57.97 
34  8.07 
30  15.43 
26  21.41 

23  22  27.37 

'            II 

-3  45.81 
3  49.90 
3  52.64 
3  54.02 

-3  54.04 

II 

-4.09 
2.74 
1.38 

-0.02 

+  1.35 

1.36 

+  1.36 

Let  it  be  required  to  express  e  in  terms  of  t,  the  latter  being 
counted  from  the  year  1000  in  terms  of  a  centuiy  as  the  unit. 

Since  we  adopt  one  century  as  the  unit  of  time,  it  is  necessary 
to  express  w  in  the  same  unit  ;    therefore  we  have 

oj  =  5  t  =   lOOOy  F(t)   =   23°  34'  8".07 


a  =   -3'  51".27   =    -231".27 
a-lc  =    -231  ".496 

Whence,  by  (366),  we  obtain 


\  =    -2".74  c  =    +1".355 

■)^li   =  50  a,'!!    =  750 


Coefficient  of     t    =    -231.496 +-      5   =    -46.299 
"  «      t'  =    -     2.74    -i-    50   =    -   0.0548 

"  "      T«  =    +     1.355 +- 750   =    +   0.00181 

Accordingly,  the  required  expression  for  the  obliquity  is  — 

£  =   23°  34'  S".07 -46".299r- 0".0548T=  +  0".00181r^ 


Verification  :  Putting  r  =^  10  in  this  formula,  we  should  get  the 
obliquity  for  2000.     Xoav  we  find 

(Per  2000)     £  =   23°  34'  8".07  -  462".99  -  5".48  +  1".81   =   23°  26'  21".41 

which  agrees  exactly  with  the  tabular  ^'alue  above. 

It  will  be  observed  that  the  solution  given  by  (.366)  restricts  the 
epoch,  or  origin  from  which  t  is  counted,  to  some  tabular  value  of  the 
argument,  as  t.  Should  the  assigned  epoch  be  some  intermediate  value 
of  T,  say   T, ,  it  Avill  only  be  necessary  to  write 


and  we  have 


T,    =     T. 


F{T)   =   F{T,  +  r,)   =   F{T,)+r,F<{T,)  +  ^^F"{2\)  + 


DR.  GEO;  ;cEWEN 

200  THE   THEOKT   AND    PKACTICE    OF   INTERPOLATION. 

Therefore,  if  we  put 

T^   =   t  +  mtsi  ~\ 

we  shall  have       j,^^^  _  f,„  + T,F:„Jr  4 F,;, +^  f::  +  ....       (      ^'^'"^"^ 

whei-c  Ti  (=  T —  T,)  is  the  value  of  the  variable  argument  counted 
from  the  assigned  epoch  T^ .  Accordingly,  if  we  compute  l)y  the 
usual  methods  the  values  of  F,n,  jF"^,  F'„^,  F',',^,  .  .  .  .  ,  and  sul)sti- 
tute  these  in   (36G('<),  Ave  shall  obtain  the  expression  required. 

As  an  example,  let  us  express  the  obliquity  (e)  as  a  function  of 
the  time  (r,)  counted  from  the  epoch  1600.0  in  terms  of  a  century  as 
the   unit. 

Reverting  to  the  above  table,  we  take 

t  =  ISOQy  T^  =    16005-  //,    =   0.20 

Whence  we  find 

F„  =  23°  29'  28".69  F',,,  =  _46".761  F;1  =  _0".0443  F'^'  =  +0".01088 

Substituting  these  values  in  the  formula  (3G6a),  we  obtain  the 
required  expression,  namely, 

£  =   23°  29'  28".69   -46".7G1t,   -0".0222Tr  +0".00181  tJ 

87.  CtEOMetiucal  Problem. — A  circular  well  four  feet  in  diameter 
is  centrally  intersected  by  a  hoi-izontal  cylindrical  shaft  whose  diameter 
is  one  foot.  Find  the  volume  of  the  portion  of  the  shaft  within  the 
well. 

Solution  :  Consitler  a  vertical  section  or  lamina  of  the  shaft 
parallel  to  its  axis,  at  a  horizontal  distance  x  from  the  latter,  and 
having  the  differential  thickness  dx.  Then,  if  we  denote  the  radii  of 
well  and  shaft  by  li  and  r,  respectively,  we  shall  have  for  the  length 
of  this  rectangular  section 

and  for  its  breadth,  or  height, 

h  =  2V?-^— ai« 


THE  THKORY  AND  PRACTICE  OF  INTERPOLATION. 

Therefore,  the  volume  of  the  differential  section  is  — 
dV  =  Ihdx  =  ^-J{B^-x^{r^-x^)dx 

V  =   ^£'J{B-~x^)(r''-x^dx 


201 


whence 


Upon  siib.stitntin<;-  tlie  g-iven  values  of  R  and  /■  in  this  formula,  it 
becomes 

r  =   8j„V(4-.r-^)a-x-^)rf.r    . 

This  expression  belongs  to  the  class  of  functions  known  as  ellip- 
tic integrals,  and  therefore  cannot  be  integrated  directly.  Accordingly, 
we  2)roceed  to  evaluate  1'  by  mechanical  quadrature.  For  this  purpose 
it  will  be  convenient  to  put 


whence 


X  =  \  sin  0 
dx  =  if  cos  6d6 


and  the  preceding  expression  for   V  becomes 


■'Ode 


(367) 


We   now   tabulate      F  (6)  =  cocos'^e  >^16—sur0      (where  w  =  10° 
77  -i-  18)   as  follows  : 


6 

'F 

F{e) 

zJ' 

J" 

A"t 

Jiv 

O 

-  15 

-  5 
+  5 

15 
25 
36 
45 
55 
65 
75 
85 
95 
+  105 

0.0000 
0.6927 
].3427 
1.9129 
2.3765 
2.7201 
2.9449 
3.0063 
3.1117 
3.1168 

0.6500 
0.6927 
0.6927 
0.6500 
0.5702 
0.4636 
0.3436 
0.2248 
0.1214 
0.0454 
0.0051 
0.0051 
0.0454 

+  427 
0 

-  427 
798 

1066 
1200 
1188 
1034 
760 

-  403 

0 
+  403 

-371 
427 
427 
371 
268 

-134 

+  12 
154 
274 
357 
403 
403 

+  357 

-  56 

0 

+  56 

103 

134 

146 

142 

120 

83 

+  46 

0 

-  46 

+  56 
56 
47 
31 

+  12 

-  4 
22 
37 
37 
46 

-46 

202  THE    TIIEOKY    AND    PKACTICE    OF   INTERPOLATION. 

Accordingly,  we  take 

<  =  5°  t  =   8  t  +  iw  =  85° 

and  proceed  by  formula  (259)  :  thus,  observing  that  Ji, ,  J'_[i,  .  ...  . 
and  J'._^.^,  z/;|'j ,  ....  are  aU  zero,  and  remenibcring  that  the  factor  w 
has  already  been  introduced,  we  find 

'F_,  =   0 

and 

V  =   '7''.+,j   =   3.11G8  cubic  feet 

88.  Various  other  problems  and  applications  of  a  similai-  nature 
might  be  added  ;  indeed.  Astronomy  itself  presents  a  large  variety  of 
such.  But  the  leading  principles  of  our  subject  have  ah'eady  been 
developed,  explained,  and  exemplified.  We  therefore  feel  confident  in 
leaving  the  student  who  has  thoroughly  mastered  these  principles, 
believing  him  fully  capable  of  solving  any  further  questions  or  prob- 
lems that  may  arise  in  his  practice. 


THE    THEORY   AND   PRACTICE    OF    nSTTERPOLATION. 


203 


EXAMPLES. 

1.  Derive  the  exjjression  for  the  sum  of  the  cubes  of  the  iii-st  r 
integers.  Ans.  \r\r-\-l)-. 

2.  Find  from  the  following-  ephemeris  the  instant  when  Autumn 
commences  ;  that  is,  the  instant  when  the  Sun's  right-ascension  (a) 
equals  twelve  hours. 


Date 
1898 

Sun's  R.A. 
a 

Date 

1898 

Sun's  R.A. 
a 

Sept.  13 
16 
19 
22 

h      in      8 

11  25  47.56 
11  36  33.99 
11  47  20.29 
11  58     6.94 

Sept.  25 

28 

Oct.      1 

4 

h       m       8 

12     8  54.44 
12  19  43.35 
12  30  34.30 
12  41  27.92 

Ans.  Sept.  22^'  12"  34'".8. 


3.     From    the    ephemeris    of    the    moon's    latitude    given    below, 
determine  the  instant  of  a-reatest  latitude  north. 


Date 
1898 


July    9.0 

9.5 

10.0 


Moon's 
Latitude 


o       /         n 

+  5     7     9.3 

5  14  28.1 

+  5  17  38.3 


Date 

1898 


Moon's 
Latitude 


July  10.5 
11.0 
11.5 


+  5  16  48.7 

6  12     9.7 

+  5     3  52.8 


Ans.  July  10"  3"  27™.4. 


4.     Given  the  equation 

sin  (s-43°)   =   0.92  sin^s 
to  determine  the  root  which  foils  in  the  second  quadrant. 


Ans.  101°  17'  43" 


5.     Given    the  following   table  of   the    longitude  of  Mercury's   as- 
cending node   (^) : 


204 


THE    TIIKOItY    AND    rUAOTICE    OF   INTERPOLATION. 


Year 

d 

1700 

44  46  34.42 

1800 

45  57  39.28 

1900 

47  8  45.40 

2000 

48  19  52.78 

2100 

49  31  1.42 

Express  6  as  a  function  of  r  ;    where  t  is  the    ehipsed  time  from 
1900,  reckoned  in  terms  of  one  century  as  the  unit. 

Ans.  e  =  47°  8'  4,5".40+4266".7.3t+0".(5307^ 


APPENDIX. 

ON  THE  SYMBOLIC  METHOD  OF  DEVELOPMENT. 

89.  While  many  of  the  fonnuhie  and  results  in  the  foregoing 
text  have  been  derived  by  somewhat  indirect  methods,  yet  the  pro- 
cesses emjiloyed  in  every  ease  have  involved  nothing  but  purely  alge- 
braic operations  and  principles. 

For  the  benefit  of  such  students  as  may  be  interested,  we  shall 
now  devote  a  brief  space  to  the  more  direct  and  potent  foi-m  of 
develojMuent  known  as  the  sipnhoUc  method.  In  this  our  only  purpose 
is  to  exhibit  the  simple  manner  in  which  the  fundamental  formulae  of 
the  text  may  be  deduced  ;  leaving  the  student  to  enter  for  himself 
upon  the  broader  field  thus  opened  by  suggestion. 

90.  Let  us  define  the  symbol  of  opfratloii  A  by  the  relation 

AF{T)   =   F{T+^)-F{T)  (368) 

from  which  we  formulate  the  following 

Definition:  The  operation  of  A  upon  any  futictiou  of  T  j>i-o- 
duces  the  increment  in  the  function  ivhich  corresponds  to  the  finite 
increment  w  in  the  variable  T. 

The  relation   (368)   may  be  more  briefly  expressed  in  the  form 

^Fn  =  ^,.+1  -  F,^  =  K  (369) 

where  n  can   have    any  value.     Thus,  taking   n  =  0,    and  referring  to 
the  schedule  on  page  15,  we  have 

/\F^  =   F,-F^  =  j;  (370) 

Similai'ly 

Ai^i  =  F„_    -  F^  ^  /l[  \ 

AF„  =  F,    —  F„  =  j:,  ( 

\  F    —    F     —  F    —    /I'  1 


206  APPENDIX. 

Thus  it  is  evident  that  the  effect  of  operating  with  A  upon  any  tabu- 
lar function  is  simply  to  form  the  fir.^t  differejice  of  that  function  and 
tlie  succccdini>-  tabular  value.     Whence  it  is  evident  that  we  have 


'o 


AAFi    =    A(J;)    =   Jy 
AAF.   =    A  (.7.')    =   z/:' 


(372) 


It  follows  that  the  operation  of  AA  upon  any  tabular  function 
produces  the  second  difierence  beai'ing  the  same  subscript.  But  this 
double  operation  of  A  may  be  conveniently  characterized  by  A-  ; 
hence  we  write 

A^i';  =  J;       ,       ^'F,  =  ^['       ,        ,       ^"-F,  =  J'J  (373) 

In  like  manner,  i  denoting  any  integer,  we  have 


A'F„  =  A(A'->F„)  -  A(J<;-")  =  4'' 
A'/;   =    ACA'-^i-;)   =    A(.'Ji'-^')    =  Ji" 


(374) 


A'/:   =    A  (A'-'/;)    =    A(J<'-")    =   z?i" 

and,  moi'e  generally,  n  being  a  non-integer, 

A'/',,  =    (AAA   ...  .   (■  times)  i^„  =  4"  (375) 

91.  Let  us  now  consider  the  operation  of  differentiating  F'(T) 
with  respect  to  T  and  multiplying  the  derivative  by  w.  Denoting  the 
operator  in  this  process  by  D,  we  then  have 


dF, 

also 


D^„   =  0,^  =   mF,:  (376) 


D'F„  =    DD/;  =   '-  —  {'-Fl)  =  u;'F:'  (377) 

D'F„  =   (DDD  ....  /  tinie,s) /''„  x=  L -^V/'',,   =   i^F^'^  (378) 

92.     The  fundamental  laws  or  i)rinciples  governing  the  combination 
of  symbols  of  quant'dij  in  algebraic  operations  are  the  following  : 


APPENDIX.  207 

I.     The  Distributive  Law,  by  vii'tue  of  which 

a  (y' +  </  +  '')   =   tip  +  «y  +  «'■ 

II.     The   Commutative  Law,  expressed  by  the  equation 

ab   =   ba 

III.     The  Lidcx  Law,  which  asserts  the  relation 

a'  X  a'   =   a'+' 

We  proceed  to  show  that  the  symbols  of  operation,  A  and  D,  when 
combined  each  with  itself  or  with  symbols  of  qnantitij  in  the  manner 
indicated  below,  also  obey  these  fundamental  laws  ;  and  hence  that, 
wherever  found  in  similar  combinations,  A  a)id  D  mat/  he  treated  alr/e- 
hraically  precisely  as  if  they  were  themselves  mere  symbols  of  quantity. 
We  shall  first  consider  the  symbol  A. 

(1).     By  definition,  we  have 

A  iF„+f„+  ....)   =   (i^„+i +/„+,+  ....)-  {F„+f„+  .    .    .    .  ) 

=  (^;+i-^„) +(/.+.-/„)+  •   •   •  • 

=    A  F,  +  A/„  +  .    .    .    . 

which  proves  the  Distributive  Law  for  the  symbol  A. 

(2)  The  factor  a  jjcing  a  constant,  we  have 

Aai';  =   aF„^,-aF,_  =   a  (i^,.+i -  i^„)   =   a/^F„ 

thus  showing  that  A  combines  with    constant   quantities  in  accordance 
with  the   Commutative  Law. 

(3)  r  and  s  denoting  positive  integers,  the  relation  (375)  gives 

A'-A'T^,,  =   A'(A'i^„)   =    A'J;."   =   J<:+"   =    A'+'F„ 

or 

A'A'  =   A'-+' 

Therefore,  so  far  as  j>ositive  integral  indices  are  concerned,  the  symbol 
A  obeys  the  Index  Law. 

93.     Retaining  the  limitations  and  the  notation  nsed  above,  similar 
results  are  easily  obtained  for  the  operator  D,  as  follows  : 


208  APPENDIX. 

(1)  D  (/-„+/„+  ....)   =   -^(^'.+/.+  .    .    .    .) 

=  "7^  +  '*';rr+  •  •  •  • 

=    Di^„+D/„+  .... 

(2)  D.,F„  =    (-/^,)<'^„   =   «-^'  =   «D^„ 

These  relations  prove  that  —  Avithiii  the  limitations  imposed  —  the  sym- 
bol  D  obeys  the  fundamental  laws  of  algebraic  combination. 

94.  To  a  limited  extent  it  is  necessary  to  consider  negative  powers 
of  A  and  D.  Now  the  meaning  and  nsc  of  A~",  A~-,  .  .  .  .  ,  and 
of  D"',  D~",  ....  are  easily  understood  :  thus,  from  the  foregoing- 
definitions,  we  have 

A('i^„)   =   F„ 

where  'F„  is  defined  as  in  the  schedule  on  page  134.  Then,  in  analogy 
with  the  usual  mode  of  expressing  inverse  functions,  Ave  may  write 

'F„  =   A-'F„ 

Whence  we  have 

A  A-'/;  =   AC/',,)   =   F„  (379) 

which  shows  (1)  that  the  operation  of  AA~'(=  A")  leaves  the  sub- 
ject function  iinaltei'ed,  and  (2)  that  negative  jwivers  of  A  also  oheij 
the  Index  Law. 

The  relation 

£^-'F„  =   'F„  (380) 

may  be  taken  as  the  definition  of  the  operator  A~'.  Similarly,  Ave 
have 

A-=i^„  =   "F„         ,         A-^i-;  =   "'F„         ,  (381) 

Again,  consider  the  relation 

D7^„  =  J^  =  V  (382) 

Avhich,  fi'om  the  ])oint  of  view  above  taken,  may  be  written 

/;  =   D-i«  (383) 


APPENDIX.  209 

Then  Ave  liaAe 


DD-'y   =    DF„  =   V 


(384) 


whence  we  see  that  negative  powers  oC  D  likcAvisc  follow  tlie  Index 
Law. 

Moreover,  from  equation   (382),  we  obtain 

dF„  =  0,-hHlT 
and  therefore 

F„    =    o,-'CrdT 

which,  with   (383),  gives 

D-i(.   =   m-'l'vdT  (.S85) 

It  follows  that  the  operation  of  D^'  is  equivalent  to  an  integration. 
More  specifically  :  Operathig  upon  ant/  function  with  D~^  integrates 
that  fmidion  with  resjject  to   T  and  divides  the  residtiiig  integral  liij  w. 

In  like  manner  we  have 

\r"-F„  =  (0-=  rfF^cZT^  (386) 

and  so  on. 

95.  Having  thus  defined  and  explained  the  use  of  the  symbols 
of  operation.  A"-,  A"',  A",  A,  A%  .  .  .  .  ,  and  0"=,  D"',  D",  D,  D%  .  .  .  .  ; 
and  having  shown  that  these  symbols  may  in  general  be  combined 
algebraically  as  if  they  were  merely  symbols  of  quantity,  we  noAV  pro- 
ceed to  dei'ive  the  fundamental  relations  of  the  text,  as  originally 
proposed. 

96.  The  theorem  of  the  change  in  sign  of  the  odd  orders  of 
differences  caused  by  inverting  a  given  series  of  functions  is  easily 
proved.  To  this  end,  let  us  suppose  that  .J;/',  of  the  direct  or  given 
series,  becomes   [j;']  when  that  series  has  been  inverted.     Then,  since 

A/^,  =  F,^,-F,  =  j; 
we  have 

-A7^,  =  F,-F,^,  =  [j;] 

Whence,  regarding  —A  as  operator,  it  follows  that 

(-A)'^^,  =  [./;■],      (-A)^7<^  =  [j;-],       .  .  .  .,      (-A)'/-^  =  [j;"] 

and  therefore 

[JS^']   =   (-A)'7^,.  =   (-l)'A"^.  =    (-l)-Jr'  (387) 

which  establishes  Theorem  III. 


210  APPENDIX 

97.     By  definition,  we  have 
hence 


A/''„   =    /'„+,  -  F„ 


(1  +  A)F„   =   F„-\-AF„   =   F,^^,   =   F{t  +  n^>  +  u,) 

=  f,^.f:^If:  +  '^f:'+.  .  .  . 

D-  D« 

=   F„  +  DF„  +  ^^  F,.  +  ^  F,,+   .    .    .    . 

=    {}+^+^+^+    ■    ■    ■    ■)  ^'.   =    «''^» 

Avhere  e  is  the  base   of   the   natural    s^'stem   of  logarithms.     AVe  have, 
therefore, 

1  +  A    =   e'>  (388) 

Avhicli  is  the  fundamental  relation  l)etween  A  and   D. 
98.     From   (388),  we  get 

0=      D'      D^ 

A=f"-l   =    D+^+l^  +  ^+....  (389) 


and  hence,  by  involution. 


A-  =    D-  +      D''+V,D'-i-TD'^+  •    ■    •    • 
A"*  =    D"  +ft  D^+  i  D^  +  :^  0-^+  .    .    .    . 

A'    =    D'   +  i  D'+'+  J^  (3i+  1)  D'+-+    .    . 


(390) 


These  expressions  are  equivalent  to  the  formulae  (21). 
Again,  from  the  last  of  (390),  we  derive 

A'F,  =   (D'  +  <i,D'+'  +  ",D'+-+  .    .    .    .)F, 
that  is 

Ji"   =   oj'F'J' +  ti,oj<+'F['+'' +  a,o>'+-Fl'+"-^  +  .    .    .    .  (391) 

where  for  brevity  we  have  written  </,,  a.^,  ....  to  denote  the  co- 
efficients of  D'+',  D'+\  ....  in  (390).  Whence,  if  F(T)  =  aT' 
^^Ti-'-\-yT'-'^  .  .  .  .  ,     we  have 

Ji"    =    a)'/*'"    =    «<u'    /4  (2")    =    u<o'[i 

(IT'  ^     ' 
which  is  the  algebraic  statement  of  Theorem  V. 


APPENDIX.  211 

i)9.     Expressing  the  relation   (388)   in  logaritlimic  Ibrni,  we  get 

D   =   log,  (1  +  A)    =    A-^+^-^+    ....  (392) 

whence 

D-   =    A--    A-''+ ]i  A"  -  ;}  A''4-    .... 

D"  =   A'-3A*+  i  A'-  .    .    .    .                                      y  (393) 


From  these  relations  the  formnlae  (45)  —  or  the  equivalent  group  (105) 
—  immediately  follow. 

100.  We  next  consider  the  question  of  reducing  the  tabular  in- 
terval from  w  to  mco,  as  discussed  in  §  19.  Since  in  the  25i'eceding 
definitions  of  A  and  D  the  magnitude  of  the  interval  is  arbitrary,  we 
have  here  only  to  denote  by  d  and  d  the  corresponding  symbols  in 
the  reduced  series  ;  evidently  tlie  same  relations  will  then  exist  be- 
tween d  and  d  as  were  found  above  for  A  and   D.      Thus  we  obtain 

d    =   »iw  -jj,  =    m  (o>  ~}j    =    >N  D  (394) 


and  since,  by  (388),  we  have 

1  +  A    =    e° 

we  must  have  also 

1  +  9   =   ed   =   e""  (395) 

AVhence  we  find 

1  +  9   =   (1  +  A)"'    =    1+  »/ A+— ^^2^ i  A-  +  — ^^ ^ '-  AH    .... 

and  therefore 

7)1  (m  —  l)       „       vi(m  —  l)(m  —  2) 
9    =    m  A  +  -^^ '-  A'+  — ^ ^ '  A'  + 

d'^  =    »rA'^+ »r(wi  — 1)  A^ \        (^qq) 

3"  =   m^A^+ 


which  are  equivalent  to  the  relations  expressed  in   (64). 
101.     The  equation 

AF„  =   F,-F^ 

may  be  written  in  the  form 

(l  +  A)/';   =   I'\  (397) 


212  APPENDIX. 

Hence  the  binomial  1  +  A  may  be  defined  as  an  operator  whose  effect 
is  to  raise  by  unity  the  sul)script  of  the  subject  function.  Whence 
we  have 

(i+A//;  =  (i  +  A)/-;  =  F,  ^       ^"■'^> 

and  generally 

(l  +  A)"7^„   =   F,,  (.199) 

We  therefore  obtain 


..  L      X    r.          /i         .      «(»  —  !)       „      n(Ti~l)(n~2)      , 
F„   =   (1  +  A)"i^„   =   fl+»A+     ^^     ^A-+-^ ^ -^A'  + 


/': 


or 

„  ^  .,      w(n-l)    ^„      n(n-l)(n-2)    „„ 

F„  =   i^„  +  «J;,+  -^-j2-^  ^i  +  -^ ^ ^"    +    .    .    .    .  (400) 

which  is  the  fundamental  formula  of  interpolation  due  to  Xewton". 

102.  We  now  find  it  convenient  to  introduce  a  new  symbol  of 
oi^eration,  which,  from  its  similarity  and  relation  to  A,  we  shall  desig- 
nate   V:     this  operator  is  defined  by  the  equation 

V/-,  =  F,  -  F,_,  =  j;_,  (401) 

From  this  relation  we  at  once  derive 

v=7':  =  vj,'_i  =  //;i„ 

v*Fi  =  vj;:'3  =  Jii, 


V3^  =  vj;:.  =  ^-  ,         ^^^^^ 


whence  it  appears  that  the  operation  of  V  upon  any  tabular  function 
produces  the  difference  of  order  /■  which  falls  upon  the  upward  in- 
clined diagonal  through  that  function  ;  whereas  the  successive  opera- 
tions of  A  produce,  as  already  shown,  those  differences  falling  upon 
the  doivnward  diagonal  line.  Moreover,  from  the  complete  similarity 
of  character  of  these  two  operators,  it  is  obvious  that  V  likewise 
follows  the  fundamental  laws  of  algebraic  combination. 

The  relation  between  V  and  A  is  easily  found  :    thus,  from  (401), 
we  obtain 

(1-V)/.;  =  F,_,  (403) 

also,  from   (307),  we  have 

(l  +  A)/',_i   =   F.  (404) 


APPENDIX.  213 

Whence  we  find 

(l  +  AXl-V)F,   =    (l  +  A)/v,    =   /'■ 

and  therefore 

1-  V    =    (l  +  A)-i  (405) 

which  gives 

log(l-V)    =    -Iog(H-A)  (406) 

Again,  combining  (388)   and  (405),  we  obtain 

1  -  V    =   e-°  (407) 

103.  As  an  immediate  apphcation  of  the  preceding  relations,  let 
lis  derive  the  formula  (75).  By  means  of  (388),  equation  (399) 
becomes 

/•„  =  (1  +  A)"/;  =  e-/; 

whence,  changing  the  sign  of  n,  we  find 

/'-,.  =  «-"°^;  =  i^-yj'^  =  (i-v)"/'; 

nfn — 1)  ^^o      n(n  —  l)(n  —  2)  __,  ,   ,, 

=    (1-"V+     ^^     W^-^ ^ ^  VH    .    .    .    .)/'\ 

Therefore 

F_„  =  F,-  nJL.  +  '^^  J-1,  -  ^^lumUlI^:)  j,^+ (408) 

which   is   Newton's    Formula  for   backward  interpolation,  as  given  by 
(75). 

104.  Formula   (60)    of   the   text    is   easily  deduced   by   means   of 
the  identity 

A    =    (1  +  A)-1 
Thus  we  find 

A'F,  =   i(l  +  A)-l\-F,     . 

=  |(i+A)' -/(i+Ar'+i^^(i+A)-^-  ....  J/-; 

whence,  by  (399),  we  obtain 

J(.)   =  F,  -  IF^_,  +  '^  /■_,  -  '^IzM^  F,_,  +    .    .    .    .  (409) 

which  is  the  same  as  equation  (66). 


214  APPENDIX. 

105.  AVe  now  paf^s  to  the  derivation  of  the  fundamental  forni- 
uhie   of  mechanical  quadrature.     Since     D  =  log  (1 -(- A),  avc   have 

f  A-      A^       A*  \"' 

D-'/'„  =  |log(l  +  A)|-'/'„=(^A-^  +  ^-^-+    .    .    .    .j     F„ 

=   (A-'+i-iVA  +  5'jA--,'^-,A'+  ,il,;A^-,;i!;S„AH    •    •    •    ■  )  F„ 

Whence,    interpreting    the   first   member    according    to   (385),  and    the 
term  A^'F,,  as  in  (o80),  we  find 

o,-'fF„dT=  'F„+iF„-^,j:  +  ^,j::  -j^^j::'+^s^j^-^^i^,j:+  .  .  .  (4io) 

This  is  the  fundamental  i-elation  of  qnadi-ature,  from  which  the 
formnla  (a)  of  (250)  is  at  once  derived.  To  obtain  {h)  of  (250)  in- 
volving the  ditferences  z/,'_, ,  .:/,','_2,  ^L'is,  .  .  .  .  ,  we  have  only  to 
employ  the  relation   (40(j),  and  the  above  development  becomes 

D-i/;  =  !iog(i+A)i-'^':,  =  s-iog(]-v)r'^''„ 

—    V  V  ^         1  J  V        i  4  V  7  J II  V  1 1;  0   V  II II 4  s  0  V  ■     •      •     •  ^  -^  11 

the  interpretation  of  which  gives 

agreeing  with  fornnda  (/>)  of  (250). 

106.  Similarly,  we  obtain  for  the  second  integration 

/  A-       A^        A''  \~= 

D-=i^„  =  siog(i+A)r=^';  =  A- :;+':-—+  ....    F„ 


=   (A-=+A-'+  ,',-,],  A-+.]nA^-^iini.  AH, ;/,^.A^-  .    .    .  )/;. 

Now   the  first   pair    of   terms    in    the    right-hand    member    may  be 
written 

(A-=  +  A-')/';   =    A-=(l  +  A)/';   =    A-=F„+,   =    "/:.+! 

and  therefore  the  pi'eceding  expression  becomes 

,.,--  C  C  F  rlT'^  —  "  F        A.  Ji     F  —      1      ,1"  J-      I      //'"_      22  1      /livi        V9      //v  _  M1'>"1 

<"        I     I^,."-'      —      ■'^i.+l  +  T^f-'^n         54lJ^'i  +  5?TJ -^.1  TrSJSO'^"  +  fflJJS  '^'i  •      •      •  \*^-) 

from  which   (324)   immediately  follows. 


APPENDIX.  215 

Again,  we  find 

D--/''„   =    Jlog(l  + A  )<-'/•',   =    >-loS(l-Vj5-=7-',. 
(  V'      V'*      V"      V"  \"'t, 

=  (v+f+f +^-+|-+  .  .  .  .)  /'; 

=  (V"— v-'+ A-auV— .].,v'-«n5ioV^-,;j^sv'-  •  •  )^:,    (4i;^) 

Transforming-  the  first  two  terms  of  the  last  expression,  wc  olj- 
tain 

(  V--  -  V-')  F„   =    v-=  (1  -  V)  /'',.   =    V  -=  ( 1  +  A )-'  F„ 

Now,  because  the  operation  of  1  -)-  A  raises  by  unity  the  subscript 
of  the  subject  function  ({^101),  it  follows  that  the  operation  of 
(l-)-A)~'  diminishes  that  subscrii)t  by  one  unit.  Accordingly,  we 
have 

(v-^-v-^/';  =  v-ni  +  A)-'/:,  =  v-=i;;_i  =  "/;+. 

and  hence  the  relation   (413)   gives 

„,-2    (       i    V  ,lTi    '//,'  J_      1      F    1       ,/"       —       1        ./'"     22'       /-/'^      19       /P  |'11J.\ 

which  is  equivalent  to  the  formula  (.326).     These  expressions  complete 
the  fundamental  relations  of  mechanical  quadrature. 


TABLES. 


218 


Table  I.  —  Xewtox's  Interpolating  Coefficients. 


3 
k 

o 

O 

< 
c 

5 

> 
-1 

la 
Q 

CO    OC     CO     t?^     (M        ^ 

^   —  I-   -*      3 

+ 

cc   ac   o   oi     >.o 

1 

00   o   CO  «re     cyD 
TH   cq   c-i  cq      oi 

C     C-1    -f    CD 
CO    CO    OO    CO 

1 

I 
s 

i 

O  CV  CO  -*  O  C-1 
C^l  1-0  CJ  C-1  »0  1- 
OC  OC  OO  C5  C5  o 
<N  C-1  N  ?l  Ol  01 

02990 
03004 
03015 
03022 
03026 

-O  CO  1-  1-  LO 

Ol  01  th  ©  C: 

©'©©'©  © 

CO  CO  CO  CO  01 

©  ©  ©  ©  © 

©  Ol  Ol  ©  -+ 

X      ©     -f     TH     C-. 

©  ©  ©  ©  X 
01  CI  Ol  01  Ol 

©  ©  ©  ©  © 

©©"*©-* 

O  CO  ©  t-  CO 
XXXI-  t- 
C-l  Ol  01  01  Ol 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

> 

S 

(N    r-   i-H    S5    o       :o 

«D    lO    »iO    ^    ■<*       CO 

1 

O    CO    rt    CD       ^ 
CO    Cq    Cq     rH        T-( 

GO    CO     <N     «^-      05 

1  + 

CO  r-  T-  Hji     t- 
th  TH  01  (M     oq 

T-    ■*   CO   O 
CO    CO    CO    ^^ 

+ 

•* 
N 

C  M  Ci  O  O  !£ 
CD  M   t-  CO   I-  tH 

I-  00  X  C5  C5  o 

CO   CO   CO   CO  CO   -f 

C  O  C'  C'  o  o 

C-l   0)  X  S5  <0 

LO    X    C    01   -!• 

C    —   1-  1-  -rt 

-*    Tf    -).    -il    -T)( 

c  o  o  o  o 

CO   -t   I-   LO  © 
LO   ©   ©  CO    CO 

5  -t  -C  -*  -f 

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tH     X      T-      ©     © 

LO   CO   01    O    l^ 

I?  i:j  rj  ^<  ^ 

©©©©'© 

©  X  -f  ©  © 

^t    rr    X    ^    C 

^   -^    Ct    CO    CO 
©'©©©© 

1                   1                1                1                1           1 

^ 

id 

^    CO    <0    C5    CM        .CO 
'-'    O    OS    CO    CO       I-- 

+  " 

00    •-'    '*    00       (N 

CD    CO    lO    •*       r)i 

>n    »    CO    CD       O 
CO    Cq    <M    TH       ^^ 

ire  cq  t~  cq     oo 

+    1 

■*    Ci    ■*    t» 

•M   cq  CO  CO 

1 

1 

e 

I 

s 

C3 

03  O  CO  05  00  o 

CO   00  00  l^  CD  LO 

-t  in  5D  t-  CO  C5 

10  lO  lO  l-O'  l-O  lO 

o  o  o  o  o  o 

LO   CO  ■*   X   CO 
CI   CI   LO  O  LO 
O  O  -T-l  01  01 
CO  CO   CO  CO  CO 

o  o  o  o  o 

X  CO  tH  -*  © 
©  CO  ©  X  © 
Ol  CO  CO  CO  -* 

©•©©©© 
©  ©  c  ©  © 

©  10  CO  ©  2 

rt  S  rt  ft  ?"' 

©5  ©  ©  © 

©  01  CO  ©  © 
t-  LO  Ol  X  '0 
CO'  CO'  CO   01  01 

CO  ©  ©  ©  © 
©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

T 

fa' 

in    1!5    lO    113    lO        i!0 
'i*    CO    Ol    ^    O       C5 
!N    <N    «■)    C^    (M       rt 

o  ».o  »r3  lO     i.o 
00   r-   CO    lO      ^ 

»[0    lO    W3    40       lO 

CO   oq    TH    o      c: 

in   ire  ire  ire     tre 

CA.    t-    CD    ire       TP 

lO    iC    lO    o 
CO     Ol     ^ 

1 

7 

CI 

lO  O  >ffl  O  LO  o 
t-  C-l  lO  00  05  o 
CO  CO  00  o  oi  »o 
Oi  C5  05  o  o  o 

O  O  O  iH  tH  T-l 

lO  O  LO  O  LO 
O  X>  lO  C-l  t^ 

CO  X  O'  01  CO' 

O    O   -TH    TH    tH 

©  lO  ©  >o  © 
CI  LO  00  ©  © 

LO  ©   I-  X  C 
^  rt  T-  TH  C-l 
f-~^   -^^   T-^   ^r^   T^ 

lO  ©  LO  ©  lO 
©  X  10  Ol  t- 
©    tH   01   CO    CO 
01  Ol  01  01  01 

7~<     T^    T^    T^    -^H 

©  LO  ©   LO  © 
Ol  LO  X  ©  © 
^  -Y  -f  ~f  i'. 
Ol  01  01  Ol  01 

1                  1               1               1               1          1 

Interv; 

ll 

s 

LO  CO  t-  (Xl  C5  O 
IM  <M  iM  M  01  CO 

tH  C<1  CO  -*  JO 
CO'  CO  CO  CO  CO 

CO  I-  X  ©  © 
CO  CO  CO  CO  ^ 

tH  Ol  CO  '*'  lO 

^  -*  ^  '^J'  ^ 

©  t^  X  ©  © 
Tt  -f  rf  "*  LO 

o 

^, 

p 

O 
b 

O 

O 

o 

e 

CD    00    C5    C^    tP       t- 
CI    00    t'    t^    CD       »r3 

+  "  " 

O     C-l     to    05        TH 
If^     -*     CO     Ol         C-l 

tH     T-t     tH     tH         tH 

CD   CJ:    '-ri    I^      O 

TH    o   O    05      c: 

tH       ^H       T-t 

ire  o  oc  CI     CO 

00    X    I-    CO       o 

X     00    'X'    -t* 

ire   ire  ^   Tf 
+ 

1 

S 

O  CO  -*  CO  LO  C-. 
O  cr>  00  CO  CO  C3 
O  iH  CO  IQ  t-  00 

o  o  o  o  o  o 
o  o  o  o  o  c 

01056 
01206 
01348 
01483 
01612 

CO  ©  X  ©  t- 
CO  Tf  IC'  CO  LO 
t-  X  ©  ©  tH 
■rt   tH   tH  01  Ol 
©'©©©© 

t-  01  Ol  10  -f 

-f  CO  tH   X    LO 
Ol  CO  -)<  T)<  lO 
IM  Ol  Ol  M  Ol 
©  ©  ©  ©  © 

t-  10  X  ©  © 

TH    I-    01    1-    01 

0  ©  I-  l^  X 

01  Ol  Ol  01  Ol 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

> 

id 

(5 

»o   t—   r*   OS   T-1      T-H 
"*    CO    (N    tH    rH       O 

C|)  oq   oq   (N   c^      o» 

T)l    lO    OO    05       IM 

05    OO    t-    CO       CO 

'^    ^-    O    (M       »C5 

JO    -*    '^    CO       (N 

05  TH  ire  05     eq 

TH      t-H     O     CS          05 

CO    O    -*    00 

CO     00     t-     CO 

1 

m 

i 

s 

O   LO  01   C5  CO  05 
O  -*  00  O  01  CO 
O  Ol  -f  I-  C-.  tH 
C  C'  O  O  O  — t 

o  o  o  o  c  o 

w  -t<  CI  t-  O 
Tf    CO  tH   05  CO 

CO  LO  t-  X  o 

^r^   T^   y~\  T^  O'X 

o  o  o  o  o 

X  <M  ©  ©  tH 
Ol  X  CI  ©  © 
<M  CO  LO  ©  X 
(M  Ol  01  01  01 

©  ©  ©  ©  © 

©  lO  ©   tH   © 
Ol  Tf  L-  CO  O 
©  ©  tH  01  CO 
01  CO  CO'  CO  CO 
c  c  ©  ©  © 

01  X  X  01  © 

10     CO    TH     05     © 
T#     lO     ©     ©     l- 

CO  CO  CO  CO  CO 
©  c  ©  ©  © 

1                 1                 1                 1                 II 

•1 

fa 
5 

00    C5    CO    O    O       O 
<N    rt    o    CO    05       O) 
CO    CO    CO   Ol    N       cq 

O    TH    cq    CO       -^ 
t'    CO    »0    Tti       CO 
M    d    cq    (N       (M 

lO    CD    t^    O       ^ 

Ol    TH    O    C5       Ci 

(M       fM       Cq       TH            TH 

oq    CO    CD   1-      O 

00  t-  CO  ire     ■* 

(M    CO    CO    CB 
^    CO    <M    TH 

"  "  "  + 

1 

O  00  t-  lO  -*  Tl* 
O  C-1  -f  >o  UO  ^ 

O  CO  CO  c:  Ol  LO 

O    ^    O    O   -r-l    ^H 

O  O  O  C'  o  o 

-*    Tl<    >0    t-   O 

Ol  c:  10  O  LO 
CO  'O  CO  CO  X 
T^  01  01  01  Ol 

c  o  ©  o  o 

Tl4  C!  'O  IM  tH 
X  ©  01  CO  CO 

©  CO  'O  t--  © 
CO  CO  CO  CO  CO 

©©'©©© 

Ol  Tjt  t-  CO  © 

01  ©  I-  -*  © 

tH  CO  -*  ©  X 

Tt     -J<     Tf      -J.     ^ 
©     ©     ©     ©     © 

©    tH    T)<    ©    © 

-cH  ©  01  LO  © 
©  ©  Ol  CO  Tl< 
-*   LO   LO   LO   LO 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

•h 

fa' 

Q 

»vO    O    >0    ».0    lO       >o 
C5    CO    1'    CD    iO       -^ 

"^   ^  "*  -^  ■*     ■* 

lo  lo  ir;  ira     lo 

CO    Cq    T-"     O       Oi 
^    "^    "^    "^       CO 

lO  ire  ira  ire     in 

00    I'    CO    lO       TP 

CO    CO    00    CO       CO 

ire  ire  ire  ire     ire 
CO  <M  TH  o     a 

CO    CO    CO    CO       C-l 

ire  ire  ire  ire 
00  i—  CD  ire 
cq  cq  cq  <M 

7 

O  »0  O  LO  O  lO 
O  Oi  00  lO  Ol  t^ 
O  •*  C-.  -t  O  CO 
O  ©  CO  1-1  T-l  C^l 

o  o  c  o  o  o 

©  LO  ©  >0  O 

(M  lO  00  O  © 
X  Ol  CO  '©  10 
CI  CO  CO  -f  -f 

c  ©  ©  c  c 

lO  ©  IQ  ©  lO 
©  X  LO  (M   t- 

X  Ol  ©  ©  CO 
Tt   LO  LO   ©   © 

©  ©  ©  ©  c 

©  iC  ©  "O  © 
IM  lO  X  ©  © 
t-  ©  CO'  ©  © 
CO  1^  1-  t^  X 

©  ©  c  ©  © 

lO  ©  «5  ©  lO 
©  X  lO  01  b- 
(M  LO  X  tH  CO 

X  X  X  ©  © 

©  ©  ©  ©  © 

*■ 

1                1                1                1                II 

InteiTal 

. 

O  1-H  M  CO  'l*  >o 

o  o  o  o  o  o 

CO  t~  00  C5  © 
©   ©   ©   ©   tH 

^  Ol  CO  -^  »o 

tH  f— 1  T— (  tH  tH 

©  t^  X  ©  © 
tH   tH  tH  1-t  01 

tH  Ol   CO  '^   LO 
<M  CI  CI  CI  CI 

o 

© 

Tablp:  I.  —  Newton's  IxTEKroLATixG  Coefficients. 


219 


O 
O 


CO  CO  OO  t-  t-  to 

C-l  O  O  -*  00  n 
•*  CO  00  01  1-1  T-i 
1— I  T— I  tH  tH  ^H  ^ 

o  o  o  o  o  o 


«D  CO  to  t^  00 

to  O  -t<  CO  C-) 

O  O  Ci  00  CO 

i-H  tH  C  O  O 

c  o  o  o  o 


C2  O  01  'f  t- 
CO  1-1  l-O  S2  M 
I,  t-  CO  10  lO 

o  o  o  o  o 
O  C'  o  o  o 


O  't  O  '^  o 

CO  C-l  CD  1-1  CO 

-f  rj(  CO  CO  (M 

o  o  o  o  o 

o  o  o  o  o 


CO  -*  (M  O  O 

o  >o  o  >c  o 

<M  iH  iH  O  O 

o  o  o  o  o 
o  o  o  o  o 


+ 


+ 


+ 


+ 


+ 


+ 


Ol    Ol    O    d 
00    00    C:    CO 


»o   :0   '^   -^ 

00    CO    00    CO 

+ 


I-  1-1  -f  t^  CO  © 

C2  ^  c-i  CO  -*  CO 

tH  ^  O  C5  CO  t- 
Ol  CI  CI  1-1  -^  ^H 

o  o  o  o  o  c 


1-1  CI  CO  ct  -f 

t^  CO  Ci  O  T-i 

O  1-0  •*  -t<  CO 

T-<    1— (  ^H  1— (  ^H 

o  o  o  o  o 


-^  'i'  UO  1.0  CO 

C<  CO'  -f  lO  CO 

CI  1-1  O  O  CO 

o  o  o  o  o 


t-  Ci  1-H   CO  CO 

t-  CZ>  O  1-1  CI 

I-  CO  CO  lo  ^ 

o  c  o  o  o 

o  o  o  o  o 


C3  -*  00  -f  O' 

CO  >o  CO  a:/  CO 

CO  CI  1-1  o  o 

o  o  o  o  o 

o  o  o  o  o 


CD    t-    CO    1^ 
CD    CO    CO    CO 


CO  C'  1-1  05  CO  o 
C  I-  CO  CO  'I'  o 
05  I-  CO  -)■  CO  CI 
CO  CO  CO  CO  CO  CO 

o  o  o  o  o  o 


C1  CO  ^  00  -* 
lO  O  10  C5  If 
O  C2  I-  lO  -i< 
CO  CI  CI  CI  CI 

o  o  o  o  o 


00  O  iH  iH  O 

a;  CO  L~  1-1  »o 

CI   .rJ   05   CO   CO 
CI   CI   iH  1-1   ^H 

o  o  o  o  o 


CO  lo  th  CO  1-1 

00  CI  CO  05  CO 
^  CO  iH  <32  CO 
■^^  iH  1— I  O  O 

o  o  o  o  o 


CO  O  CO  t-  o 
CO  O  CO  CO  C' 

CO  10  CO  1-1  o 

o  o  o  o  o 
o  o  o  o  o 


+ 


+ 


+ 


+ 


+ 


lO    lO    »0    »0    iC       iA         >0    lO    lO    »0 


»0    to    Its    ifS 

<©  t-  CO  c:! 

CO    CO    CO    CO 


"*      Tt*      -S*      tP 


■^    tJ*    ^^     ^ 


+ 


lO  O  lO  c  >o  o 
t-  CI  10  CO  C5  o 
CO  tH  00  LO'  CI  o 
C5  O  CO  00  00  CO 

o  o  o  o  o  o 


LO  o  >o  o  >o 

C5  CO  lO  CI  t- 
CO  CO  o  t--  CO 
t-  t^  t-  CO  CO 

O  C  C  C'  o 


O  no  O  lO  o 
CI  lO  CO  Ci  o 
O  CO  CI  CO  lO 
CO  lO  LO  ^  -* 
O  O  O  C'  C: 


lO  O'  lO  O  10 
C5  CO  lO  CI  1- 

C  CO  CI  00  CO 

'a-  CO  00  CI  CI 

o  o  o  c  o 


O  1-0  O  lO  o 

CI  lO  00  C5  C: 

C3  -sjt  C5  -^  C' 

1— i  1— I  O  O  w 

O  CO  o  o  c 


Interval 


lO  CO  l- 

t^  I-  I- 

<o 


CO  c;  o 
t-  t^  CO 


1-1  CI  CO  -*  lO 
CO'  CO  OO  GO  CO 


COt^COOiO         iHCICO'^LO 
COCOCOCOO         Oi  C.  Oi  a  c. 


CO  t-  CO  05  o 

Ci  o  o  c;  o 


00    GO    O     ^H 
-t    •*    lO    o 


I'    1^    Oi    00 

lO    o   o   to 


o  o  o  o 

CO    CO    CO    CO 


-*  CO  t^  LO  CI  CO 
CO  C5  LO  1-1  t-  CI 
t^  CO  CO  CO  'O  lO 
CI  CI  CI  CI  CI  CI 

o  o  o  o  o  o 


CI  't  CO  CO  LO 
CO  CO  00  CO  CO 

-#  -*  CO  CO  CI 

CI  CI  CI  CI  CI 

O  O  O  O  C' 


CO  O  lO  1-1  »o 
CO  00  CI  t-  1-1 
CI  iH  iH  O  O 
CI  CI  CI  CI  CI 

o  o  o  o  o 


CO  rt   -*  LO   t^ 

LO  O  -^  CO  CI 
C5  C5  CO  t-  t- 
1— I  tH  ^H  1— I  ^H 

o  o  o  o  o 


CO  00  CO  CO  CO 
o  CO  -(<  oo  CI 

O  CO  lO  -*  ■* 
1— I  iH  rH  1-^   tH 

o  o  o  o  o 


+ 


+ 


+ 


+ 


+ 


+ 


o 
o 

"1 


o 

3 


CO    CD    CO    C^ 
Tt<    If    1*     ITS 


+ 


00    •-<    CO    lO 
O    CO    o    ^ 


c;  o  CO  "* 


O    --I 
00     00 


CO    CO     lO    ^ 
00    CO    CO    iXi 


+ 


CO  CO  I-  c;  I-  -* 

O   CO  '^   O  ^H   CO 
05  00  CO   I-   t-  CO 

CO  CO  CO  CO  CO  CO 

o  o  o  o  o  o 


1-  cv  :o  LO  o 

CD  't  CO  CI  CO 
CO  lo  -t*  -*  CO 

CO  CO  CO  CO  CO 

o  o  o  o  o 


CO  -^^  -*  1-1   t^ 

c:  CI  LO  00  o 
CI  CI  -rH  o  o 
CO  CO  CO  CO  CO 

o  o  o  o  o 


CI  LO  t^  I-  CO 

CO  lO  t^  Ci  1-1 
Ci  CO  t-  O  CO 
CI  CI  CI  CI  CI 

o  o  o  o  o 


-t<  iH  CO  CO  I- 
CO  lO  CO'  CO  C5 
10  -+  CO  CI  1-1 
CI  CI  CI  CI  CI 

O  O  C'  o  o 


(M  t-  O  lO   OS 
t-  t-  00  00   00 


O  CO  I-  CO  -ct<  tH 

LO  O  LO  O  -f  ^' 
CI  CI  ^<  1-1  O  C5 
CO  CO  CO  CO  CO  lO 

o  o  o  o  o  o 


Tt  CI  lO  LO  o 

1-1  -t<  CO  CO  o 

en  00  t-  CO  CO 
lO  10  10  LO  »o 

O  O  O  'O  c 


^rH   02  CI  CI  C5 

^M  1-1  CI  CI  1-1 

10  — t^  CO  CI  tH 

10  lO  lO  10'  l-O 

O  C'  o  o  o 


CI  iM  t^  o  o 

T^  'O  00   I-  LO 

O  Ol  t-  CO  lO 

LO  -)<   -t-   -t"  ^ 

o  o  o  o  o 


l^  iH  CI  O   CO 

CI  O  fr-  -*  o 

-*  CO  1-1  O  C5 

^  -r(<    -).    -H    CO 

O  O  O  C'  o 


+ 


+ 


+ 


+ 


+ 


+ 


»0    itO    lO    1ft 

?D    t-    QO    05 


+ 


O  '-I     (N     CO    '^ 

CM  CI     CM     C^    <M 


+ 


O  LO  o  >o  O  lO 
O  05  00  lO  CI  l- 
LO  -H  -*  -*  ^  CO 
CI  CI  CI  CI  CI  CI 


O  LO  O  I'O  o 
CI  lO  CO'  C2  o 
CO  CI  1-1  O  'O 
CI  CI  CI  CI  CI 


LO  O  LO  C  LO 
C5  CO  LO  CI  l^ 
00   t-   CO  LO  CO 


O  >0  O  LO  o 

CI  lO  00  03  o 

CI  O  00  CO  o 

iH  1-1  o  o  o 


LO  O  l-O  O  lO 
CI  00  LO  CI  t^ 

CI  O  CO  CO  00 

C;  O  Ci  Ci  C3 
1-1  tH  C'  o  c 


Interval 


O— IC1CO-*L0         COt-OOOiO         tHCICO'^iO  cot-cocao         ■^CICO'^LO 

lOlOLOLOlClO         LOLOlOLOCO         OCOCOCOCO         COCOCOCOl-         t—t-t-t—  t- 

d d 


220 


Table  II.  —  Stirling's  Interpolating  Coefficients. 


o 
O 


(M       ^    ^    c;    o 


t-     t^     (^     tT         -f  CO     C-l     r-l     rt         — 


+ 


+ 


05  T*   Ci  CO   t^  O 

to  05  t-i  "*  ^  o: 
I-  t-  00  OC  00  oo 
O  O  O  O  O  O 

o  o  o  c;  o  o 


<M  CC  '^  ec  C^l 

T-l   CC  L^    t^  O 

C3  Oi  C:  o:  c: 

o  o  o  c  c 

o  o  o  o  c 


^-1  00  lO  C  lO 
vH  <M  -J.  tC  t- 

o  c  c  c  o 


CI  CI  ^  o  o 

CO  O  T-(  CJ  CO 


>0  CO  O  t-  CI 
-Tf  1.0  O  O  l^ 


occc       cocoo 


+ 


+ 


+ 


+ 


+ 


rt<  CO  CI  ■rt  rH  ^ 
<*  O  00  O  CI  Tit 
C)  CI  CI  CO  CO  CO 

o  o  o  o  o  o 
o  o  o  o  o  o 


CI  CO  •*  »  X' 

:o  cc  O  CI  -T 

CO  CO  -*  -*  'j* 

O  O'  C'  o  o 

o  o  o  o  o 


o  c^  o  t~  c 

t-  C5  T-J  CO   CO 
1<  ^  l-O  1.0  lO 

c  o  o  o  o 
c  o  o  o  o 


CO  lO  CC  O  CO 
OO  CO  LO  t- 

LO  ?r  CO  tt  to 

C   C'  C'  O  CT' 
o  c  c  o  «=• 


1-0  I-  C5  S  1-1 

c.  1-1  ?t  CO  00 

^0  t~  I-  I-  t^ 

O  O  O  C'  o 
O  C'  o  o  c 


O    t~    (M    t- 

00  r-  r-  o 


CO  O  CI  T-H  t-  O 

O  -f  I-  O  CI  o 

03  C  ^^  CO'  -i*  LO 

00  tC  ^  -c  -*  -* 

o  o  o  o  o  o 


O  t-  tH  CI  C5 
I-  CO  C  <-^  -n 
iO    I-    CrS   C'    rH 

-*  -^  ^  O  i.O 
o  o  o  o  o 


C)  CI  C!  tH  O 
CI  CI  1-1  tH  O' 
CI  CO  ^  LO  CO 
O  lO  IQ  'O  »0 

o  o  o  o  o 


».0'  lO  CI  ^  tH 
OO  CD  'd*  1-1  00 

CO  i^  X)  c;  C5 

LO  O  »0  lO  o 

o  o  o  o  o 


^  CO  t-  CO  o 

^  O  lO  C  lO 
O  -rt  rt  CI  CI 

CO  CO  CO  CO  CO 
o  o  o  o  o 


lO  O  O  lO  Ift 
ITS  CC  t—  00  O 
CN    (N    <M    ff^    C^ 


+ 


CO  CO    CO    CO    CO 


»0    lO    iC    o 
CO    t-    30    Oi 

CO    CO    DO    CO 


>f3     iC     ^ 
^     tM     CO 


O     »C3     to     >0 

CO  r^  CO  OS 

■^     "^     "^     ^^ 

+ 


lO  O  lO  o  o  o 
CI  <»  -t<  CI  o  o 
1-1  CO  CO  c;  CI  "O 

CO  CO  CO  CO  '^  -t< 

o  o  o  o  o  o 


lO  C  LO  C  i-O 

O  CI  -f  CO  CI 

CO  -^   -+   l^  1-1 

-f  lO  l.O  l-   CO 

o  c  o  c  o 


O  LO  O  lO  o 
CO  •*  CI  o  c 
-*  CO  c>  -o  o 
CO  CO  i—  t-  X) 
o  o  o  o  o 


LO  O  «I  O  lO 

C   CI  -C  CO  C] 

-f   CO  CI    CO  iH 

(Z)  OO  C-.  c:  'O 
O  O  O  C  1-1 


O  lO  o  »o  © 

CO  -*■  CI  o  o 

'-0'  O  KO  O  LO 

C    1-1  IH  C)  CJ 


+ 


+ 


+ 


+ 


+ 


+ 


Interval 


lO  CO  [~  CO  C:  O 
CI  CI  M  CI  CI  CO 


i-(  CI  CO  '^  1-0 

CO  CO  CO  CO  CO 


CO  t^  00  03  o 
CO  CO  CO  CO  ^ 


T-l  C5  CO  1*  l-O 

*^3^     ^^     ^^    ^^    ^^ 


O  t-  00  c^  o 
^  ^  ^  ^  iO 


+ 


+ 


O  CO  t-  O  CO  CO  GC  CI  1-0  I-  O 

OCOCOOCOCO  C3COCOC5C1 

O  O  O  rt  -H  1-1  -H  C)  CI  CI  CO 

oooooo  ooooo 

ooooco  ooooo 


tH  CO  -)<  lO  CO  CO  CO  CO  1-0  Til 

CDOCllOt/3  1-lTfl-OCO 

COCO-*-*-t<  lOlOlOCOCD 

ooooo  ooooo 

ooooo  ooooo 


CI  OJ  t~  CO  C5 

CO  00  ■rt  ^  CO 

CO   CO   t^  I-  I- 

ooooo 
ooooo 


+ 


+ 


+ 


+ 


+ 


+ 


o* 


O    C^    (M    CO    CO       in 


OCDt-GO       CI         OiOi-HC^       C^ 


CO    -^    -t^    »f3       CD 


CD    1—    t—    00 


O  O  CI  ^  I-  o 
O  O  O  C  C  rt 
O  O  O  O  O'  o 

oooooo 
oooooo 


^^^ 

,  ---- 

"  "  "  T 

10  O  CO'  CO  1-^ 
1-H  CI   CI  ot.  -f 

ooooo 

OOOOO 

ooooo 

O  O  C-.  C   CI 
i-O  10   O   X   3-. 

ooooo 
ooooo 
ooooo 

^  [-  1-^  1-0  o 

O'  ■rH   CO'   rt'   CO 

1— '    T— <    T-<    1— 1    1— * 

ooooo 
ooooo 

CO  CI  c:  O  '* 
[^  C5  O  CI  -* 
■^  1-1  CI  CI  CI 

ooooo 
ooooo 

t-CDI:-CDlO       1(5         ira-*CO(M       »-l         00300CD       -*         COC^OSOO       O         c0C^O5CO 
OCDCOCDCO       CD         CDCDCDCO       CD         COiOlOlO       lO         i.TtO'^-*       rl^         iJ"i*COCO 


O  t-  CO  O  CO  tH 
O  CO  CO  C  CO  CO 
C  1-1  CO'  1-0'  CO  cc 

oooooo 

OOOOOO 


CO    r-l    l-O   X   O 

O  CO  CI  'X  lO 

o  1—1  CO  -f  CO 

O  tH   1— (  1— *   tH 

OOOOO' 


iH  iH   O  X  -1< 

iH  I-  CO  X  -* 

X  o  1-1  C)  -f 

1-1  1-1  CI  CI  CI 

Ooooo 


X  1-1  CO  CI  o 
.C3  1-0  O  lO  o 
10  I-  O  O  CI 
CI  CI  CI  CO  CO 
OOOOO 


CO  05  1-1  O  CO 
•*  CO  CO  I-  O 
CO  '^  CO  t-  05 
CO  00  CO  CO  CO 

ooooo 


CD  I-  a;   c: 


iO    »^    »A    o 
«D    t-    00    Cs 


+ 


O        rH    M    CO    "* 


+ 


"=l« 


O  10  O  lO  O  l-O 

O  O  CI  -*  X  CI 
O  O  O  O  O  iH 
OOOOOO 
OOOOOO 


O  1-0  O  lO  o 
OO  -t  C)  o  o 
-^  CI  CO  -t  10 

ooooo 
ooooo 


lO  O  lO  O  lO 
O  CI  T)<  00  CI 

CO  t--  X  C5  1-1 
O  O  O  O  1-1 

ooooo 


O  lO  O  lO  o 
00  -^  C)  O  o 
CI  -^  CO  00  o 
iH  tH  iH  iH  CI 

ooooo 


lo  ©  ira  o  lo 

©  CI  »*  00  CI 
CI  ■*  CO  00  iH 
CI  CI  CI  CI  CO 

ooooo 


+ 


+ 


+ 


+ 


+ 


+ 


Interval 


O  iH  CI  CO  ■*  lO 

ooooo© 
o 


O  t-  X  o  © 
©  ©  O  O  1-1 


tH  CI  CO  -f  10 


'  CO  C5  © 
I  1-1  iH  C5 


tH  CI  CO  Tfl  lO 
CI  CI  CI  CI  CI 


DR 


TauL?:    II.  —  STIKLTXfj's    TxTERPOLATIXfi    COEFFICIENTS. 


221 


El, 

M 
O 
O 


CO 


-f    50    I-    CO    Ci        — 
(M    C^     G-l     (M     C-\        CC 


O  50  O  CO  l«  O 

-+   -^   C5  O  C-O    O 

C5  r>  X  yj  cc  X 
o  o  o  o  s  s 

o  o  o  o  o  o 


O  >0  C-l  35  ^ 

I-     -Tt     1-1     l_--     -J 

o  5  5  o  o 


+ 


CO  0-1  -*  lO  lO 

O  t^  CO  C^  'O 

w  o  »o  -t>  -t< 

o  o  C;  O  o 

o  s  o  o  o 

+  '  '  '  ' 


CO  rt  00  -C  CS 
— I   I-  o  CC  CO 

-t  CO  CO  c-1  :■  1 
o  o  o  o  o 
o  o  o  o  o 

+  '  '  '  ' 


CO  O  30  o  o 
05  -*  C5  10  O 
tH  -J  O  O  O 

o  c  o  o  — 
o  o  s  s  o 


+ 


+ 


+ 


^     (M     W     CC 


O    '^i'    t-    rH 
O    t-    t-    00 

+ 


»0  t-  to  CO  t-  o 
N  iH  C  C5  I--  O 
O  O  O  C-.  C:  C5 

— ,  ^  -^  o  o  o 
o  o  c  o  o  o 


O  00  CO  to  »o 

-t<  iH   C5  to  CO 

o  05  CO  cC'  » 


c-1  t-  00  to  tH 

O  CO  iM  CO  -* 

2?  t  —  '""'  '~ 

o  o  5  o  o 


CO  (M  t-  C5  t^ 

OJ  -^  CO  CI  to 
"O  O  -1"  -+  CO 

o  c  c  o  o 
o  o  o  o  o 


T-l  C-l  00  IH  © 

O  CO  >o  cc  o 
CO  C-I  »-^  o  o 
c  c  o  o  o 
O  o  o  c;  O 


I 


+ 


t'    CO     »0    tJ^        CO 

o  w  (M  CO     -r 

(W     C^     (M     C-J        ^1 


'M'-'OO     o      Oia>ooo 

o  -i:?  1-  oj      c:      cs  o  --H  (7) 

C"l     CI     CM     7-1        C*l  CM     O?    CO    CO 


+ 


en  O  -f  1-1  o  o 

to  lO  C)  03  -^  o 

■^  CO  IM  O  C:  OO 

O  10  »0  lO'  -^  — t' 

O  O  O  O  O'  c 


CO    t-    Tjl    (M    tH 

-*  t-  O  IM  CO 

to  ^  CO  -rt  Oi 


(M  LO  CV  Tf  c: 

CO  C-l  O  t»  IQ 

t-  10  CO  'O  CO 

CO  CO  CO  CO  CI 

O  C  O  O  C' 


t-  lO  -*  T)<  ^ 

O  lO  C3  (M  -* 

to  CO  O  OO  lO 

CI    CI   CI  1-J   -rH 

o  o  c  c  o 


■*  lO  t~  CO  c 

10  10  ^  CI  o 

CI  CJ  to  CO  c 
▼-<  O  ■—  O'  o 
C  ex   C:    O  C 


+ 


ic:i   iQ  ifo     ifo      lo  «o  ICO  >o 

I--OOCi        O  — i(MCOTt< 

1—    t-    t-        X  iX    CO  .  CO    CO 


liO  ifO  iO  »o 
O  t-  CO  Oi 
CO    CO    CO    CO 


lO    tO    o    o 

--■    <M    OO    -^ 

Oi    Ol    Ol    c& 


c:   Oi   C:    c; 


+ 


lO  O  LO  O  lO  o 
CI  00  ^  CI  o  o 
— '  OO  to  -*(  CI  o 
00  CO  C3  O  -rH  CI 
CI  CI  CI  CO  CO  CO 


lO  O  10  C  lO 
O  CI  •*  CO  CI 
CO  to  -*  CI  T-l 
CI  CO  -*  lO  to 
CO  CO  CO  CO  CO 


C:  lO  S  lO  O 

CO  -t<  CI  o  c; 
05  00  t^  to  lo 

O  l~  00  C3  o 
00  00  CO  CO  -t 


lO  O  lO  O  10 

—■  C)  -^  CO  CI 

-t  CO  CI  tH  »-l 

■-I  CI  CO'  "*  lO 

^~i*  ^^  ''^  ^^  '^' 


C  lO  O  lO  c 

a;  -t<  CI  o  c 

o  o  o  c  o 

to  t^  CO  Ci  o 

^    •*  -)<    -rf    lO 


+ 


+ 


+ 


+ 


+ 


+ 


Interval 


lO  to  I-  CO  C5  o 
t^  t-  t-  I-  t-  OO 


^  CI  CO  -t<  lO 
CO  CO  CO  CO  CO 


to  t-  QC  c;  o 

CO  CO  OO  00  c: 


—I  CI  CO  -f  lO 
C:  O  C--  O  CI 


to  t-~  OO  Ci  c 

C:  C5  Ci'  C5  O 


o 
o 


-*     Ot)    -T-l     .-H 


00   c:    O    Ol 


+ 


+ 


CI  to  05  tH  CI  CI 
t^  t-  It-  CO  OO  CO 


T— I  Ci  lO  O  lO 

cc  t-  t-  t-  to 


00  O  iH  T-i  Ci 

LO   lO    ^    CO   rH 


to  CO  CO  CI  ^ 

c  o  t-  to  -t 

T-H-Hi— l-i— i^H-— ^  -r^^HiH^H-r^  i-t  T-*   t^   T-t   t~*  -i-H'OOOO 

T— '^Hi— I^Hi— (1— ^  ,— (,—(,— «,—(,—(  ,— (,— (^Hi— *^^  tHt— (*— It— '1— * 

oooooo   oooso   ooooo   ocooo 

+ 


to  to  lO  CO  o 

CI  C  CO  to  -)< 

O   O    'CS   C3  C3 

^H  T-(  ex  o  o 
C=  c  ©■  c  s 


+ 


+ 


+ 


+ 


+ 


i-    CD    'O    -J'       -+ 


c-i    ^    ^ 


I--    iCO    OO    lOI        i-H 


C)    CO    O    t- 


I    + 


+ 


tH  CI  CM  (M  T-l  C5 

OC  O  CI  -i<  to  t~ 

I-  OO  OO  CO  CO  CO 

C'  O  O  C'  o  o 

o  o  o  o  o  o 


t-  'S*  O  to  C' 

o;  1-1  CO  -t<  o 

CO   C5   CI  C3  O 

O  O  C'  O  C: 


-*<  to  t-  00  t- 

t-  OO  Oi  ©  T-i 

©  C:  ©  O  © 

©    O  O  rH   -H 

©  ©  © 


^     T-l    to    05    T-l 

CI  CO'  CO  CO  -* 
©  ©  ©  ©  © 


©       ©  ©  o  c  © 


CI  ©  t^  CI  lO 

^  -1<  CO  CO'  CI 

©  ©  ©  ©  © 

T-*  ^H  r-i   ^H  tH 

©  ©  ©  ©  ■© 


I 


O    -*    Oi    -t*    CO       o^ 

OO    CO     C-t     CM     T-i        .-< 


+ 


+ 


©  ©1  CO  CI  to  -* 
l-O  00  CI  lO  l^  C3 

CI  CI  CO  CO  CO  CO 

to  to  to  ©  to  to 
©©©©■©© 


to  CO  uo  ©  © 

©  tH   tH  tH  © 
^   ^   ^   *^   "^ 

to  to  to  ©  © 
©  ©  ©  ©  © 


-t<   T-l   CO  C»  to 

CO  ©  CO  ©  lO 
CO   CO   CO   CI  CI 

to  ©  '©  ©  © 

©  ©  ©  ©  © 


OO  -*  CO  LO  © 
©  lO  ©  CI  LO 
CI  r-l  ©  ©  © 
©  ©  ©  ©  lO 
'C  ©'  ©  ©  © 


00  ©  CO  ©  © 
©  t~  oC'  CO  © 
oc  t-  ©  10  -^ 

IC  LO  10  lO  lO 

©©©©'© 


iO  to  O  O  >Q 
O  ^  C^  CO  ■'t 
lO    O    >0    >0    lO 


+ 


O    O     lO    lO 

o  r-  CO   C5 

iCT'    ifO    ifO    »iO 


»0   O    lO   »c      uo 

,-.     (M    00    -^        O 

CO    CO    CO    ^       CO 


iCO    ifO    lO    t.'O       »f3 

CO   i^   CO    OS      c: 
CO    CD    CO    O       t- 


1-  t-  r-  r- 


+ 


©  lO  ©  lO  ©  lO  ©  1-0  ©  lO  ©  lO  ©  LO  ©  LO 

©  ©  C)  ■*  00  CI  OD  -t<  CI  ©  ©  ©  CI  -+  CO  CI 

LO  ©  LO  ©  10  TH  ©  CI  00  •*  '©  ©  CI  CO  -f  -^ 

CI   CO  CO   -*   ^   LO  LO   ©  ©  t-   CO  oc    ©  ©  ©  "-1 

T— (tHtHtHt-It— I  1— It— ItH^Ht-*  i— It-Ht— (CICl 


©   10   ©   LO  ©' 

CO  '^  CI  ©  © 

1-  -Jl  ^H  C»  LO 
—I  CI  CO  CO  'Td 
CI  CI  CI  CI  CI 


LO  ©  lO  ©  LO 

©  CI  -t<  CO  CI 

C)  ©  ■©  CO  "H 
LO  lO  ©  I-  OO 
CI  CI  CI  CI  CI 


+ 


+ 


+ 


+ 


+ 


Interval 


©  -rH   CI   CO  '^   lO 
lO  lO  LO  lOl  LO   LO 


©  t-  CO  ©  © 

LO  lO  lO  lO  © 


rH  CI  CO  ^  LO 

©  ©  ©  ©  © 


©  t-  OO  ©  © 

©  ©  ©  ©  l- 


tH  CI  CO  -^  >0 

t-  t-  t-  l^  t- 


222 


Table  III.  —  Bessel's  Intekpolating  Coefficients. 


o 

b. 
H 
\A 

E 

O 

w 

m 

W 
25 

> 

-1 

to 

5 

t-"    ^    ^1    ^    tM       C^ 

05  <M  CO  -^     c:) 

re    «    Tf    t       -t 

10     'i-     -t     LO        -^ 

i.O    lO    f    o 

+ 

r 
+ 

S 

1.0  -*  CO  1-1  C-.  I- 

OO  00  00  X'  I-  1- 

o  o  o  o  o  o 
o  o  c  o  o  c: 
o  c  o  o  o  o 

0  CI  O  t-  CO 
l^  C^  t-  CO  CO 

©  ©  o  o  © 
©  ©  c  ©  © 
©  ©  ©  ©  © 

©  ©  CO  ©  LO 
©  1.0  lO  -*  ■<* 
©  c  ©  ©  © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

00041 
00036 
00032 
00028 
O0023 

©  -*   ©   LO  © 
IH   -H   ©  ©   © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

1                  1               1               1               II 

> 

5 

OS    O    iCi    M    (M       d 
-^     '^     Tji     -^     ^         CO 

+ 

CO  o   -^   ^     o 

CO    CO    CO    CO       CO 

GO    CO    CO    Ol       O 
CI    d    (N    C^       Ol 

J-    CO    CO    CI       Oi 

r-  CO  CO   rt 

+ 

I 

t 

S 

05  00  '^  05  (N  ■* 

C  >0  O  -*  05  CO 
I-   t-  OO  CO   CC   05 
T-t   1-H   7— 1    1-H    ^^   -rH 

o  c  o  o  s  o 

CO  1-1  CO  ©  tH 

t-  i-(  ^  CO  1-1 

©  ©  ©  ©  -^ 

1-i  CI  CI  CI  CI 

o  ©  ©  ©  © 

1-1  ©  1.0  00  © 
'J"   ©  ©   iH   ^ 
1-1   M  iH   CI   CI 
CI  CI  CI  CI  CI 

©  ©  ©  ©  © 

©  I-  CO  ©  oo 

©  t-  ©  ©  1-1 

CI  CI  C-l  CO  CO 
CI  CI  CI  CI  CI 

©  c  ©  ©  © 

t-  ^  ©  CO  "* 
C<l  CO  '*  -*  -^ 
CO  CO  CO  CO  CO 
CI  CI  CI  CI  CI 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+          + 

T 

5 

-H     -t     t,     GO     rt        CC 
^     rH     rt     ^      71         C^l 

1 

-t    t-    CO    Ci        -H 
<M    <M    CI    (M       7Z 

CO    M*    lO     CO        t- 
CO    CO    CO    CO       CO 

00    c»   o    c:       O 
CO    CO    ir    rt       M- 

^    CI    ^    CI 
-t     TT     ^     rt 

1 
1 

s 

to 

iH  O  O  C3  »-(  O 
00  I-  l-O  CO  <M   O 

I-  t^  t-  t-  t-  t- 

c  c  o  o  o  o 

O  O'  o  o  o  o 

+ 

I-  CO  ©  00  05 
I-  lO  C-1  Oi  CO 

CO  ©  CO  iQ  lO 
©  o  ©  ©  © 

+  '  '  "  ' 

CO  lO  -rH  ©  © 
CO  ©  t^  COi  © 
lO  lO  •*  Tt<  -* 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

CO  lO  ©  ©  © 

©  CI  CO  'Ti"  © 

CO  CO  CI  CI  CI 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

+  '  '  '  ' 

©  110  CO  CI  © 

©  CI  CO  '^  © 
rt  ^  ©  ©  © 
©  ©  ©  ©  © 

©  ©  c  ©  © 

+  '  '+  ■ 

T 

id 

5 

»0    lO    to    O    »0       lO 
-^    CO   cq    »-<   O      OS 
^1    M    (N    7^    (N       T-l 

1.0    UO    lO    »c       o 

CO  r-  CO   o     Tt" 

>0    lO    If5    lO       lO 
CO    *M    1-1    O       c: 

o  o  >f^  ico     »ro 

C»    t-    CO    o       ■* 

io  iO  kfi  in 

CO    G^     ^ 

1 

J 

©» 

»n  o  lo  o  'c  o 

t^  C^J  IC  oo  o  o 

CO  5C  00  O  C-l  l-O 
C5   C5  C5  O  C    O 
O  ©  O   T-l  1-1  •rH 

lO  ©  lO  ©  lO 

©  00  lo  CI  t^ 

©  CO  ©  CI  CO 

©  ©  1-1  1-^  1-1 

©  lO  ©  lO  © 
CI  lO  CO  ©  © 
lO  ©  l-  CO  © 

1—    1—   T^   ■rH   CI 

LO  ©  lO  ©  LO 
©  00  lO  CI  t^ 
©  iH  CI  CO  CO 
CI  CI  CI  CI  CI 

1— (  1— (  1— 1  1— i  1— 1 

©  LO   ©  LO  © 
C)  LO   CO   ©  © 
^  Tji  -*  ^  lO 
C^l  CI  CI  CI  CI 

1— 1  1— I  iH  iH  1— 1 

1                  1               1               1           ■     1          1 

Interval         s 

lO  O  I-  CO  Ci  o 
C-l  CI  11  CI  CI  CO 

1-1  CI  CO  -*  lO 

CO  CO  CO  CO  CO 

©  t~  00  ©  © 
CO  CO  CO  CO  -c 

iH  CI  CO  ^   lO 

©  t^  00  ©  © 
1*    Tt    -^    Tt<    U5 

— 

© 

a 
o 
t. 

g 

•A 
O 

O 
cc 

» 

H 

C3 

> 

in 

5 

O)  CO  r-  i^  o     I- 

1 

ir:     »C    kO    O       rji 

CO    ^    CO    Ol       CI 

CI    CI    ^    o       ^ 

1      1 

O     O     -H    —1 

+  + 

I 
1 

1 

O  CO  O  CO  O  'O 
•  O  C  i-l  CI  CO  CO 

o  o  o  o  o  o 

O  S'  C'  o  o  o 
O  C'  o  o  o  c 

...... 

CO  00  00  CO  CO 
-)<  -*  UO  10  © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 
©©■©©© 

1 

-.00067 
.00070 
.00074 
.00077 
.00079 

iH   CO  LO.  ©  © 
CO  CO  03  00  00 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

©  c  ©  ©  © 

..... 

I-   t^   t-  ©  LO 

00  CC  CO  CO  oo 

©  ©  ©  ©  © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

..... 

> 

(5 

CO     (M     1-t    O     CI        OD 
00    X    CO    'X    1--      t- 

+ 

t'    CO    -*     -t       iM 

1-  I-  1-  1'     t— 

O    O    00    CO       o 

r-  t-  CO  CO     CO 

't    CI    O    Ol       t- 

co  CO  CO   o     iro 

CO    -*    CI    o 
O    >0    »£0    >0 

1 

s 

O  CO  »c  tc  -o  o 
C-  CO  O  ^  CI  o 
O  O  iH  C)  CO  '^ 
O   C:  O  O  O  O 

c  c  o  o  o  c 

CO  ©  ©  ©  r)< 
l»  ©  CO  -rt  00 
-*  10  ©  t-  t- 

©  ©  ©  ©  © 

©  ©  ©  ©  © 

©  ©  ©  ^  © 

lO  CJ  ©  ©  CO 

CO  ©  ©  ©  1-1 

©   ©  ©  iM  1-1 

c  ©  ©  ©  © 

lO    ©   1-1    tH   © 

©  10  CI  00  ^ 

1-1  CI  CO  CO  Tf 
r-*  1-H  1— '  ^H  1— 1 

©  ©  ©  ©  © 

t^  CO  t-  ©  © 

©  lO  ©  lO  © 
-*   LO   ©  ©   t- 

iH    1— t  1— 1   1— (  1— 1 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

T 

5 

^    CO    •-<    «0    (M       GO 

00  t-  t~  CO  to     ira 

CO    C»    lO    O       CO 
»o    ^    ■*    -I'       CO 

CO  00  iO   oi     a» 

CO    Ol    CI    CI       ^ 

-t*    '-H    X    i.'T'       CI 

+ 

1-1     t     t-    05 

1        1 

T 
I 

to 

O  iH  t-  OO  -*  CO 
O  CO  iO  CI  05  o 
O  O  tH  CI  CI  CO 

o  o  o  o  o  o 
o  o  o  o  o  o 

-*  t^  lO  ©  © 
iH  ©  ^   ©  © 

•i<  -f  iTj  1.0  © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

©    ©    t-    CI    Tf 

CO  ©  ©  CI  Tf 

©  ©  ©  t-  t~ 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

CI  ©  t--  lO  © 

©   I-  00   ©  © 

t~  t-  t-  t-  CO 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

CI     T-l     t-    ©    1-1 

©  ©  ©  ©  00 
00  a;  t-  i^  t^ 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+          + 

•i 

*3 

5 

»0    tC    ira    »0    »C5       lO 

OS    00    C'    CO    lO       '^ 
^    tfi    r)<    r)i    ■*       -)< 

ira  »o  m  »to     wo 

CO    N    1-1    O       <J> 
-*     ^     Tf     -t         CO 

iQ    to    lO    lO       iCi 

00    t-    CO    «5       '^ 

CO    CO    CC    CO       CO 

UO    l£0    O    tra       >0 

CO  CI  ^   o     o: 

CO    CO    CO    CO       CI 

in  o  110  lO 
O)  t-  CD  ira 

CI    CI    CI    CI 

T 

d 

O  "C  O  >.0  O  "O 
O  C5  X  1-0  CI  l- 

c  ■*  c;  ^  Ci  CO 

O  O  O  T-l  ,-1  CI 

o  o  o  o  o  o 

©  1.0  ©  >o  © 
CI  lO  CO  05  © 
00  C)  ©  ©  10 

CI  CO  CO  -+  -* 

©  ©  ©  ©  © 

1.0  ©  1.0  ©  lO 
©  CO  10  CI  t- 

CO  CI  ©  ©  CO 
'I'  lO  lO  ©  © 

©  ©  ©  ©  © 

©  lO  ©  lO  © 
CI  LO  00  ©  © 

I-  ©  CO  ©  © 

©  t^  t-  t-  CO 

©  ©  ©  ©  © 

10  ©  lO  C  LO 
©  CO  lO  CI  t- 
CI  10  00  ■r-  CO 

oo  00  oo  ©  © 

©  ©  ©  ©  © 

1                1                1                1                II 

Interval 

s 

O  iH  CI  CO  ^  lO 

o  o  o  c  ©  o 

©  I-  CO  ©  © 
©  ©  ©  ©  tH 

1-1  CI  CO  -*  lO 
^H  1— t  1— 1  iH  iH 

©  I-  CO  ©  © 

tH   1— (    1— *    1-H   CI 

■r^   CI    CO    -+    LO 

CI  CI  CI  CI  CI 

© 

=> 

Table  III.  —  Bessel's  Ixterpolating  Coefficients. 


223 


£ 

H 
•A 

W 

(d 

O 

r, 

X 

> 

5 

+  +    *  7  "' 

^     -M    cq    CI        -M 

1 

CO    -1"    CO    Tl<       ,.0 

1.0    lO    iO    I-       CO 

i-  r-   X   CO 

1 

I 

% 

5 

o 

10  50  l^  t^  I-  O 

CO  a>  CO  Gc  cc  oc 

o  c  c  o  o  o 

O  O  O  O  O  O' 

o  o  C'  =;  o  o 

CO  in  CO  th  C5 

CO'  CO  CO  00  I- 

O  O  C'  c  o 
O  O'  o  o  o 
o  o  o  o  o 

t-  "ct*  ©  t-  CO 

t-  I-  t-  CO  CO 

©  ©  ©  ©  © 
©  ©  o  ©  © 
©  ©  ©  ©  © 

00  CO  00  CO  CO 

in  in  •*  -*  CO 

©  ©  ©  ©  © 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

©  CO  ©  CO  © 
CO  Cl  1-1  ©  © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 
c  ©  ©  c  © 

+ 

4- 

+ 

+ 

+       + 

> 

la 
(5 

O     CI     -i^     CD     t-        OS 

lO  iQ   t.t.   ir:   o      o 

1 

o   cq   "*   1»     o 

O    O    CD    CD       CO 

CO    O    O    IM       •* 
CD    t-    It-    t-       1^ 

-+    CO    (-    CO       Cl 

I-  t-  t-  r-     I— 

O    "    <N    CO 
CO    CO    CO    00 

1 

i 
+ 

S 

C3  O  f-  CO  t-  o 
C:  ».■;.  O  '0  OS  -^ 
1--  -O  O  'O  -1"  -)< 

T-*   rH  ^4  T-H  T-H   i-H 

O  O  O  O  O'  o 

th  1-1  CI  in  o 
CO  CI  in  C5  CO 

CO   CO   CI    rH  T-H 
tH   ^^   ^H   1-H   1— i 

O  O  C  C'  o 

-+  O  CO  CO  'cfi 

CD  ©  Cl  in  CO 

©  C-.  05  O)  I- 

S§i§i 

©  CO  ©  CO  in 

1-1  CO  CO  CO  © 

I-  CO  m  -f  >* 
©  ©  ©  ©  © 

©  ©  ©  ©  © 

©  ©  in  CO  © 

Cl  -*  ©  CO  © 

CO  Cl  tH   ©   © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+       + 

T 

5 

O     t-    -t^     ^     ?!        O 

1              1  + 

X    ^    rt*    CC       -M 

^     ^     r-.         (M 

lO    00    CO    CO       o 
(M    (M    CO    CO       -* 

.0    00    00    CO       Ol 

Tti    -t    LO    lO       CO 

^3    I-    I^    CO 

+ 

T 

s 

X 

T 

to 

T-l  O  t^  1-1  C-l  o 
00  OS  05  o  o  o 
l^  I-  t-  OD  CO  CO 

o  o  o  o  o  c 

c  o  o  o  o  o 

in  t-  CO  CI  -t 

O  00  l^  O  "* 

t-  t-  t-  t-  t^ 

o  c  c  o  o 
o  C'  o  c;  o 

Cl  t-  O  CO  © 

Cl  C5  CO  CO  © 

t-^  CO  CD  CO  CD 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

©  in  t-  T*  CD 

©  -rH  ©  1-i   in 

in  i.n  -t<  -*  CO 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

-t  CO   t-  tH  © 
©  Cl  O  CO  © 
Cl  Cl  1-1  ©  © 

©  ©  ©  ©  © 
c  ©  ©  o  © 

1                   1                1                1                II 

^ 

lei 

i5 

o    lO   o    o    lO      »:^ 
O    CO   IT-   00   O       O 

iri  <N  jq  (N  (N     CO 

»o  »o   »o   »A      o 

•-H    (M    CO    -i'       O 
CO    CO    CO    CO       CO 

»o    1:0   o    iC      lO 
CO    t—   00   OS       o 

CO    CO    CO    CO       tP 

lo  »a  >o  lo     »o 

«    <M    CO    T*       O 

^'       ■^        ^        "^               "^ 

>0    i.O    lO    to 

CO    1-    00    Ci 

Tt       Tji       i#      1^ 

+ 

7 

c» 

O  O  lO  O  l-O  o 

t--  CI  in  CO  Ci  o 
ct  th  CO  'O  CI  o 

C2  c:  00  CO  CO  CO 

O  C'  o  o  c  o 

in  o  in  o  m 
ci  CO  m  CI  I- 

CO  CO  O  I-  CO 

1-  t^  1-  CO  CO 

o  o  o  o  © 

©  in  ©  LO  © 

Cl  O  CO  ©  © 

©  CO  Cl  00  in 
CD  in  in  -*  -f 
©  ©  ©  ©  © 

in  ©  o  ©  in 
©  CO  in  Cl  t- 

©  ©  Cl  OC  CO 
-t<  CO  CO  Cl  Cl 

©  ©  ©  ©  © 

©  in  ©  in  © 
Cl  in  00  ©  © 
©-*©-*© 

iH  1-1   ©  ©  © 

©  ©  ©  ©  © 

1                   1                1                1                II 

Interval 

s 

lO  O  I-  CO  c^  o 
t^  I-  t-  t-  t-  CO 

^  CI  CO  -*  in 

CO  CO  CO  CO  CO 

CD  t-  00  ©  © 
CO  CO  CO  CO  © 

th  Cl  CO  Ti(  in 
©  ©  ©  ©  © 

©  l^  00  ©  © 

©  ©  ©  ©  © 

o 

1— ( 

O 
t. 

-/. 

H 
?; 
H 

O 

(d 
« 

> 

(5 

1^    -^    >0    lO    ^       iO 

+ 

Til       -^      O       rtl            -t 

Tti    CO    -t    CO       -rP 

CO    Ol    CO    01       Ol 

Cl     e-l     rH     rt 

+ 

1 
+ 

o 

O  lO  CM  'f  Oi  CO 
O  O  O  -rH  tH  CI 
O  O  O  C'  c  o 

o  o  o  o  c  o 

c»  CI  CO  T-i  in 

C)  CO  CO  T*  -+ 

©  ©  ©  ©  © 
©  ©  ©  ©  © 
©  ©  o  ©  © 

©  CO  CO  ©  CO 

^  in  in  CO  © 
©  ©  ©  ©  © 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

1-  ©  Cl  lO  t~ 
CO   t^  t-   t^   t- 

©  ©  ©  ©  © 

©  ©  ©  ©  © 

©  ©  ©  ©  © 

©  ^^  CO  -^  in 

t-  00  CO  CO  00 

©  ©  ©  ©  © 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

+ 

+ 

+ 

4- 

+                + 

> 

5 

i-H     C--:     CO    I-    C5        (M 

1 

«    CO   t-   o      « 

CO    CO    00    O       — ' 
Ol    IM    Ol    CO       CO 

-i'    »c^    c»   O      Ol 
CO    CO    CO    CO       Tt 

CO    \Ci    CO    Cr. 

-*     Tt<      -t      ■* 

1 

+ 

^  CO   C  -^<  t-  CO 

-t"  -t>  -^  CO  ci  — 1 

CO  CO  CO  CO  CO  CO 
C)  CI  CI  CI  CI  CI 

o  o  ©  o  o  o 

CO  CO  t^  ©  o 

©   C3   1--  CO   Ti< 
CO  CI  CI  CI  C1 
CI  CI  CI  CI  CI 

©  ©  ©  ©  © 

02218 
02195 
021(39 
02141 
02111 

©     ©    T-l    CO    ^ 

CO  T*  1-1  1-  CO 
©  ©  ©  ©  © 

Cl  Cl  Cl  iH  1-1 

©  ©  ©  ©  © 

Cl  ©  "*  00  © 

©  ^  ©  in  © 
CO  CO  CO  t^  t^ 

1— 1  -rH   1— 1  iH  1— 1 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

T 

la 

5 

C-J    ^    (M    ^    O       O 

^  "*   -f  -t   't     'l^ 

O    OS    CO    I^       CD 
'H    CO    00    CO       CO 

lO     -*     CO     r-<         Cl 
CO    UO    0?    CO       C>l 

CO    t-    -W    CO       ^ 
C^l    !M    Ol    Ol       Ol 

00  r-  -f   i-H 

""7 

1 

X 

V. 

C  CI  CO  "S  o  o 

O  -*  GO  CI  'O  o 
O  C  O  tH  ^  CI 

o  o  o  —  o  o 
o  o  c  o  o  o 

■  1 

CO  CO  m  CO  © 

^  OD  CI  CO  © 

CI  CI  CO  CO  -f 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

1    '   '   '   ' 

CO  th  m  CO  © 

CO  t-  ©  CO  CO 

-*  -)<  in  in  in 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

1 

OO  ©  CO  t-  © 

©  Cl  in  t--  © 

in  ©  ©  ©  t-- 
©  ©  ©  ©  © 
©  ©  ©  ©  © 

1 

1-1  ©  ©  ©  1-1 
Cl  CO  in  t-  00 
I-  t^  i^  t-  t- 

©  ©  ©  ©  © 
©  ©  o  ©  © 

..... 

n 

la 

5 

iO  »C5  lO  »ra  lO     xd 

T^  <s\  vi  ^     »a 

+ 

K5    lO    lO    lO       lO 

CO    t—    00    35       O 
>— 1 

lO  >ra  »fO  to     lO 

^   c-i   CO  -p     If: 

lo  in  lo  >o     lo 

CO    t—    00    Oi       o 
r-.     rt     ^     ,-1         Cl 

lO    t.O   to   in 

iH     Cl     CO     -t 
Cl    Cl    Cl    Cl 

7 

ei 

o  o  o  o  o  »o 

O  C5  CO  O  CI  t- 
»0  ^  -t  Tf  -f  CO 
CI  CI  CI  CI  CI  CI 

©  in  ©  in  © 
ci  in  00  C3  © 

CO  Cl  i-H  ©  © 
CI  Cl  Cl  Cl  Cl 

T^  r-<  -r^   T^   -^^ 

in  ©  in  ©  in 
©  CO  m  Cl  t- 
00  t~  CO  in  CO 

©  in  ©  in  © 
Cl  in  CO  ©  © 
Cl  ©  CO  CO  in 

iH   rH  ©  ©   © 

in  ©  in  ©  in 
©  (x>  in  Cl  t-- 

ci  ©  CO  ©  CO 
©  ©  ©  ©  © 

1-1  -rH  ©   ©  © 

1                  1               1               1               1          1 

Interval 

O  tH  CI  CO  "*  1.0 

lo  lo  in  lo  o  lo 

CO  t^  CO  Ci  © 

in  in  in  in  CO 

1-1  Cl  CO  Ml  in 
CO  CO  CO  CO  CO 

©  i^  CO  ©  © 

CO   ©  CO  CO  t- 

1-1  Cl  CO  T)<  in 
t-  t-  t-  t-  t- 

o 

© 

224 


Table  IV.  —  1*^ewton's  Coefficients  for  F'{T). 


1 

U, 

<A 

3 

fa 
fa 

O 

> 
1 

5 

c;i--f(MO     CO      oocoff^     o      G  a  <X)  zrj     ao 

-rOIC-I-iO        CO         t-mo^        CO         (>)005QO       1- 
T-t-^Tj*-*       CO         COCCCOCO       CO         COO?(M(N       <N 

CO    O    -#    M       C-)         -H    «    C    C-. 
C^CMO)(N       (M         CM(NC^— ' 

1 

-40 
+ 

\ 

"sis 

T— liM»Ci-HCaCi         THiCOt-»0         iOlO'^C^IO 
CO  CO  ^  CM  O  O         C>1  -f  OC  <M  OO         -*  O  t-  t-  CO 
iTH  O  C-1  GC  -t*  O          ^  M  CC  L*  -^          tH  *t   1-  o   ^ 
Tj-  ;■■:  00  C-I  ~1  C-I         1-1  --I  O  O'  O         O  O  C:  1—  T- 

OOOCJOC'         OOOOO         OOCCO 

OO  lO  C5  O  OO         CO  LO  LO  CO  CO 
COOCO-J-)"          L-C500C1 
OOi-t-CCO         COOOOLOCO 
-^  --  CI  CI  CI          CI  CO  CO  CO  00 

OOOOO         OOOOO 

+                                           +                       +            1                                        1                                        II 

> 

5 

OC5CSOO     CO      1-t^r-i— 

oocoin-p     Q7      w— 'oos 

COCOCOCO       CO         C0«C?C^1 

+ 

1 
1:1" 

C  -*   C-.  t-  CO   »          -*  -H  C5  C»  O          C^l  t-  CI  -^  CC 

■-H-f'X-Cr-O          C:S-rH-rrC5          '1<0(»COCO 
lO  C-.  ^t  CO  CC  00         CI  03  CO  CZ)  00         O  O  C^  CO  l- 

50  >-■:  1-  -f  -t  oo       CO  c-1  CI  rJ  -H       OOOOO 

OOCOOO         OOOOO         OOOOO 

LO  LO  ^  CO  iH         C~.  CO  CO  O  t- 
Cl  O  I-  CO  00         1-1  -C  CO  I-  CO 
—  10  00  CI  lO         C5  CI  LO  CO  1-1 

-H   -H  tH  CI  CI           CI  CO.   CO  CO   ^ 
OO'OOO          OOOOO 

1                  1               11  + 

+               +         + 

'I 

fa' 
Q 

»o  iO   in  lO   ira     o      1.-^  U5   lO  o     lo      o  lo  io   >ra     ».o 

-^COOl-HO       O;         GOf-OiGt       -f         COC<1-hO       CS 
t-t-t-t-t-       CO         OOCOO       CO         OCDCOCO       >n 

i 

+ 

T 

COOOOOOOCOOO         OOOOOOCOOO         COCO)COO0CO 
1(5  -H  I—  L-O  OO  00         CO  lO  l^  i-<  O         -H  l~  l-O;  CO  CO 
-*  I-  O  CI  lO  CO         tH  -It  t-  rt  -ti          CO  -tH  LO  C5  CO 
-H  O   C3  02  CO   t-          t--  CO  lO  LO  -*          CO  CO  C1  -H   — I 
-H-HOCOO         OOOOO          OOOOO 

00  CO  CI  t-  CI         t-  CI  t^  M  t^ 

CO  lO  CI  CO  -i<         CO  C5  ^  CO  CO 
t--H-tC:LO         OCOiHCOi-i 
OOOO-H         CICIOOOOrtt 

OOO.  C.O         OOOOO 

+                +              + 

+  +  1           1           1 

n 

'.0'*C0C<I-r-4O          OCOt-OlO         -^COC^T-iO 
C-1  CI  CI  C?  0-1  CI         -H  1-1  I-*  tH  r-l         i-H  iH  iH  tH  rH 

d 

1 

CSC<3t~COL0         -^COCIi-IO 
OOOOO         OOOOO 

<6  <6 

1 

Interv 

il 

s 

LO-Ot-COCiO         i-IC^ICO-i+LO         Ol-COClO 
CI  CI  CI  CI  CI  CO         CO  CO  CO  CO  CO         CO  00  00  CO  ^ 

T-i  !M  cc  -t  i.'^       :r>  I-  cc  c:  o 

^ 

o 

Coefficients  for 

> 

(5 

i/;t-ocot-     o      ^co^co     1-1      iiroioo     lo 

OlOOit-iO       rjt         vNOCrit-       CO         tPCO^O       CO 
CCOOt-t-l^       1-         t-l^-COCO       CO         OOCDCO       lO 

I-H    t-    CO    Ol       \Ci         CI    CC    lO    CM 

1 

+ 

1 

OOOOCOLOO)        CO-tOluOCJi        cococooooo 
O   1-  O  I-  O  -t<         O  CO  t^  CO  O         ^  O  I-  »o  l-O 
O-HCOl-OOCO         COlOCWiHlO         OOCIlOCiOO 
O'  O  CO  t-  CO  -O         1.0  -*  CO  CO  M         -H  1-1  O  C5  es 
Cli-Hi-liH-^rH           T-.,H--liHiH           lH^i-(00 

CO  CI  lo  CI  CO       a;  CO.  CO  CO  1-1 

I-  O  •*  O  l-         "0  lO  CO  05  CO 
t^  CI   CO  .r^  >0          O  LO  O   10  -^ 
GO.  CZ)   1-   t-   CO           CO   LO  LO   -f  -f 

OOOOO         OOOOO 

+                +              + 

+               +          + 

T 

<a 
Q 

c-ij*o»oo     CO       ^-r-cC'X     ifT      —  r— -coo     co 

OOCOCOiO       CO         T-IOClt-       CO         l.'^COO^l-^       C5 

cicoxccx     CO      cocot-t—     t—      t-i-r-t-     co 

+ 

-+♦ 
1 
a 

+ 

s 

1 

O  -^  1-  t-  CI  CI         CO  LO.  CO  LO  t-         C-1  iH  -IH  tH  -H 

OOCi-Hio.  O         CO-HCO-ct<CO         OL0t-.C5C» 
OOrHCO-fCO           l^ClTHCOiO           CZ;OCO»-OGO 
lO    -t<   CO  CI   -H   O           C3  CO    CO   1^  CO           LO.  lO  -1<   CO.  CI 
CI  CI   CI   C)   CI  .C)           T-l   --   -H    iH  -rt           1-^  -H   -H   -^  -H 

1 1 1 

LO  CI  00  t^  CO         CO  CO  ^  o  o 
C»  O  CO.  t-  CO          O  00  00  O.  -H 
-^  LO  CO  -^  LO         CS  C)  CO  O  LO 
CI  ^  O  O  CS         CO.  CO.  t-  1  -  -o 

„-H  —  tHO.        ooo.oo 

1 1   ■  ■  '  l" 

■H 

la 
5 

in  »o  lo  in  tn     in      in  »n  o  »n     in      in  in  in  >n     in 
c:3ooi—  coin     -^      cotNi-io     o      ooi—  coo     ** 

C505005CS       CS         <35CSOSOS       CO         OOOOOOCO       CO 

in  in  in  in     m      in  in  in  m 

CO<Ni-<0       CS         QOX-COm 
coxoooo      t-       t-t-t-t- 

1 

+ 

T 

COCOOOCOCOOO         COOOCOOOCO         0500030000 
CO  CO  "C  1-  i-H  LO          --(  I-  LO  CO  CO          CO  "0   t^  tH  lO 
COCOCOCO-it-t         lOlOCOt-OO         OiO--lCO-# 
CO  CI  1-1  O   C5  CO          I-  CO  LO  ^  OO         C-1  CI  -H  O  C5 
CO  CO  CO  CO  CI  CI         CI  CI  CI  CI  CI         CI  CI  CI  CI  rH 

COCO.  COOOCO         «5C0C0C0CO 
1-1  I-  lO  CO  CO         CO  LO  1-  1-H  lO 
CO  t^  C5  T-(  CO         lO  I-  CS  CI  -t 
CO  t^  CO  CO  LO         T*'  CO  CI  CI  1-1 

+                +              + 

+               +         + 

n 

O  O  OO  I-  CO  LO         'd*  CO  CI  i-(  O         C5  00  t-  CO  >o 
LO  -!<-*-♦<-)<-(<          ^n^-*"*--)*          COCOCOOOCO 

-*  CO.  CI  1-J  O         CS  00  1-  CO  10 
CO  CO  CO  00  CO         CI  CI  CI  CI  CI 

d                                                                       ■    ■ 

1 

o 

1 

Interval 

~ 

o-ncico-fio       coi-a;c50       i-(Cicoi*>o 

OOOOOO         OOOO-H         -HT-li-li-lT-i 

COt-QOCSO         i-IClCO-fO 
T-i  1-1  1-1  T-l  CI         CI  CI  C)  CI  CI 

d 

o 

Table  IV.  —  jSI"ewton's  Cokffioients  for  F'(T). 


225 


« 
o 
u. 

(n 

h 
Z 
Ed 

»-« 

It 

o 
O 

> 

6: 

a 

»-H    O    O    O    OS       o 

7  1    +     " 

O    CO    X    C^       CO 
»-<    C^    C^    CO       CO 

O    -P    1^    71       t-O 

T    i^'    't    lO       ..O 

■X     Ol    i.O     x       — 

tro    CC    CO    w      I- 

-r   CO   o   01 
I-  I-    :c    X 

+ 

+ 

"si" 
1 

1-0   O  "H   tH   CO  t~ 
C)  CO  -f  --f  CO  C-I 

o  o  o  o  o  o 

CO  to  CO  to  CO  «o 

o  o  o  o  o  o 

C^l  CO  O  CI  o 

tH   C3   I-  -(<  ^ 

O  C3  C5  C-.  o 
CD  O  iCi  O  i.O 
O  O'  o  o  o 

-*    -*    O    CO   ■rH 
l~  CO   C3  -*  C5 
00  00  I-  I-  CO 
lO  10  LO  10  >o 

o  o  o  o  o 

■O  X  CO  1-1  CO 
CO  t-  1-H  lO  X 

CO  10  10  ^  CO 
lO  10  LO  LO  o 

O  O  O  O  'O 

05312 
05238 
05162 
05082 
05000 

1                 1              1              1              1          1 

5 

CI     CM     TJH     I—     w         M 

O    CO    lO    '^    Tp       cc 

+ 

+   1 

CD    CO    O    lO       rt 
-H    C<1    CI       CO 

x   CO   x  in     05 

CC    T     11'    iC       lO 

CO     C     CO     -H 
CD    t-    t-    X 

1 

r 

T 

"el" 

^  CO  "O  O  CO  t- 
C5  O  (M  t-  CI  CO 

>o  CO  t-  t-  CO  00 

00  CO  00  OO  CC  CO 
O  O  O  O  O  O 

O  CO  CO  OO  l-O 
O  CI  -*  O  CD 
C5  0-.   C3  O   C5 

00  00  CO  CO  20 
o  o  o  o  o 

^  CO  LO  IC  o 
CO  LO'  'SI  CI  o 

C5   C5   05  O  CS 

00  CO  00  00  00 
o  o  o  o  o 

Ci  1^  X  O  LO 
CO  CO  X  Tf  X 
X  X  t-  t^  CD 
X  X  X  X  X 
O  O  O  O  O 

08626 
08560 
08490 
08414 
08333 

+ 

+ 

+ 

+ 

+          + 

^ 

5 

lO     »0    lO     iO     O        lO 

-ii    CC     C-1     '-'     O        C5 
CI     <M     CI     (M     (N        ^ 

1 

lO    lO    O    1(0       o 

00    t-    CO    <0       'i- 

IC    lO    >0    »f3        »fi 

CO    C<I    i-H    O       C^i 

O    itO    »f3    If3       ifS 
X    C-    CO    lO       -iji 

lO    >ra    m    lO 
CO    <M    ^ 

1 

+ 

s 

1 

=8h 

CI  t-  0^1  t^  CI  t- 

^  00  CI  ^  CO  CO 
lO  t-^  O  CI  -*  CO 
CO  00  -*-+-*  -* 

CI  (^  CI  t-  cq 

CO  ^  CI  OO  ^ 

CO  C'  CI  CO  lO 
"*    LO   UO  LO  <0 

T— <    1— 1    T— 1     1— *    1— 1 

t^  CI  t-  CI  l^ 

00  CI   ■5)4   CO  CO 
CO  00  C3  O   -^ 
lO  LO  lO  CO  CO 

^^    T-*    -r^     T-^    tH 

CI  l^  CI  t^  CI 

CO  -f  CI  X  rf 
CI  CO  -t<  -i-  LO 

CO  CO  CD  CO  CO 
tH   1—*  1—t   1—1   1—f 

t^  CI  t-  CI  t- 

X  CI  -I-  CO  CO' 
LO  CD  O  CO  CO 
CO  CD  CD  CO  CD 
^  ^H  rH   1— <  1— ( 

1                  1               1               1               1          1 

T 

_l 

»C.  CO  t^  CO  C-.  o 
CI  CI  CI  CI  C)  CO 

1-1   CI   CO   ^   LO 
CO  CO  CO  CO  CO 

CO  l^  CO'  C-.  o 
CO  CO  CO  CO'  Tf 

^  C-1  CO  I*  LO 

'^  1*  -*  -*  -* 

CO  t-  X  C5  o 

^   ^   -Tt<   -t  LO 

o 

+ 

o 

+ 

Interval 

s 

1.0   CO   l^  GO  C5  O 

t-  t-  t^  i^  I-  CO 

1-H  CI    CO   ^   LO 
CO  CO  X  00  iM 

CO    t^  CO  C5   O 

CO  CO  CO'  CC'  ci 

T-<  d  CO  T)i  lO 

35  C2  CC  C-.  c; 

CO  t-  X  c;  o 
C5  C:  cr:  CC  O 

o 

1- 

o 

b 

Eh 

Z 

a 
3 

b 

w 
o 

0 

CO    lO    CO    OO    02       ^ 

ao  1^  CO   ic  ^     ^ 

^    CO    CC    -H       -* 
CO    (M    — *     i-c       O 

CO    O    (N    CO      o 
O    CS    X    1^       t' 

Ol    I^    C    -^        X 
CO    o    »n    -^       CO 

CO    t-    —    I.O 
CO    CT    Ol    ^ 

1 

1 

+ 

4 

"sU 

1 

"sis 

CO  iH   CD  CI  O  C3 
05  00  lO  C5  CO  CI 
CD  CO  O  CI  CO  'O 
CO  CO  -*  -^  rt<  ■* 

O  o  o  o  o  o 

1-1  lO  T-i  C5  o 
t-  O  CO  -*  CO 
CO  CO  O  O'  1-1 
-*   rf  -f   LO   LO 
O  O  O  O  O 

^  O  O  CI  CO 
CO  CO  LO  CO  O 

CI  CO  ^  LO  CO 

10  10  0*0  10 

O  O  O  O  O 

X  o  t-  t-  -^ 

l^  ^   C5  ^    C5 

CO  t-  i~  X  X 

LO    LO    iO   LO   lO 

o  o  o  c  o 

C5  CI  C5  O  LO 
CI  CO  X  1^  CI 

C5  o  cr.  o  o 

LO  10  10   CD   CD 

1                  1               1               1               1          1 

> 

5 

CO    I-     t-     I-    CO        CO 

CO  r-  o   .o   >*      CO 

(M     (N    N     <M     CJ        (M 

+ 

c::    O   O   ^      CI 

0  oi   -1  o     en 

01  C^    C^     C<I       T-l 

CO  »r;   CO  I-     05 

X     t-    CO     ^        -rf 

C     Ol    1*     ..O       X 

-f    CO    Ol    ^       o 

O    Ol    -1^    t' 

O    O    X    f- 

+ 

T 

+ 
1 

"sla 

t-  CO  O  I *  CI 

CO  la  CO  C3  "O  C' 

1-1  '-)<  t^  C5  Cl  lO 
^  ^  -i<  Tti  lO  lO 

o  o  o  o  o  o 

o  crs  o  C5  o 

-*  CO'  CO  C:  O 

l^  C5  1-4   CO   CO 
LO  lO  CO  CO  CO 

o  o  o  o  o 

CI  lO  O  CO  CO 

C-.    t-   LO   1-4    I- 

t-   C5  1-,   CO   -t< 

CO  CO  r-  1-  i^ 

O  O  O'  o  o 

07622 
07762 
07894 
08018 
08133 

1-1  1--   CO    t )" 

-*  -f  CO  -^  C: 

'CI    CO   irf    LO    LO' 

X  X  X  X  X 

O  O'  O  O  'O 

+ 

+ 

+ 

+ 

+         + 

n 

!ri 

5 

o   m   o   o   o      lO 

Oi     GO    I^    CO     O        "^ 

Tt^     "l*      ''4*     ^^      Tf          ^3* 

»!0    lf3    tr5    uo       o 
CO    C^    T^    O       05 
^    "^    -"^    -^       CO 

o  LO   >ra  o     lio 

O)    t-    CO    LO       -^ 

CO    CO    CO    CO       CO 

»0    «0    wO    to       lO 

CO    CM    —     O       ~. 
CO    CO    CO    CO       Ol 

to    O    to    o 

X     t-    CO     i^ 
CI     (N     CI     (N 

1 

+ 

T 

t-  M  t-  CI  l~  C<I 

CO  CO  ■*  CI  CO  -^ 
i-H  CO  1-1  CO  o  o 
-*  -H  lO  lO  CO  CO 

o  o  o  o  o  o 

t~  C<1  t-  CI  t- 

CO  CI  rf  CO  CD 

era  Tt  CO  CI  o 
CO  t^  i^  CO  00 
o  o  o  o  o 

CI  t-  C^  L^  CI 

CO  't  CI  00  -* 

O  Tt<  CO  rt  LO 

C5   C2  OS  O   O 
O  O  O  rt  1-^ 

1-  CI  t-  CI  t- 

X  C)  -f  CO  CO 
X  C«  LO  X  1-1 

O     •TH     1-      -H     CI 

CI  t-  CI  t-  CI 

O  ^  CI  x  -t< 
-f  l^  O  C)  LO 
CI  CI  CO  CO  CO 

1                  1               1               1               1          1 

^ 

O  -H  0-1  CO  -*  >0 
O  O  O  O  o  o 

CO  t-  00  C5  O 

O  O  O  O  1-f 

^  C-1  CO  '^  LO 
1— t  1— 1  tH  1— 1  1— 1 

CD  t-  X  c;  o 

1— 1    TH    1— 1    T-1    CI 

^  CI  CO  TjH  LO 

d  CI  CI  CI  CI 

o  o 

+ 

o 

+ 

Interval 

s 

O  ^  CI  CO  -*  o 
»0'  IC  O  »C  LO  iO 

CD  It-  CO  C5  O 
LOi  LO  »0  LO    CO 

tH  C)   CO  -I*   LO 

CO  CO  CO  CO  CO 

CO  t-  X  C5  o 
CO  CD  CD  CO  t- 

tH  CI  CO  t  IC' 

t-  t-  t-  t^  t- 

o 

- 

226 


Table  V.  —  Stirling's  Coefficients  for  F'{T). 


trj 


so 


■^    o    I—   c: 

C-1 

«    »«    C-    X 

o 

C'l    tt 

t'   1-   I-   I- 

00 

CO    00    CO    00 

03 

ro  =: 

+ 


00  I-  ^  C5  CI  J1 

50   O   Tt    t-  rt    --*■ 

o  "n  •*  o  0-;  01 

CI  I-l  CI  CI  CI  CI 

o  o  o  o  o  o 


tH   t-  ■rH  Tf    1,-5 

I-  35  CI  ':»<  ^ 

l-^  O  O  O  X 

CI  CI  CI  rt  >-H 

c:  c  o  s  o 


cooioxc       ccc'soe^ 


CO  O  T-H  CI 
I-  L-  '~C'  «  -:)< 


lO  »~   O  I-  I-- 

CO  CJ   rt  C'  C3 

7-1  ^^  ^H   1-H  O 

O  O  O  C'  o 


in  o  1-0  en  OS 

t^  t-  l~  l~  o 

00  t-  «>  o  ^ 

O  O  O  O'  o 
O  O  O  C'  o 


+ 


+ 


+ 


+ 


4- 


+ 


-t  o  t-  -^ 


M  ^        T-^ 


CC    C:    to        ^ 


+ 


+ 


1 


M-*Clt-00  t--T-l-.H.XC1  ClOid-rHt-  XlOXt-i-H 


CI  I-  CI 

X  X  C^  ~  C    C: 

th  1-1  i-H   -^  CI  CI 

o  o  c  o  o  s 


X  CI  l.~  1^  o 

C'    ^-^    T-l    ^    C) 

CI  CI  CI  CI   CI 


CI  CO  lO  CO  o 
CI  CI  CI  M  CI 
CI  CI  CI  CJ  CI 

O  O  'O  o  o 


CO  CO  o  '^  CO 

C)  CI  CI  Cvl  CI 

M  C)  CI  CI  C^l 

o  o  o  o  c 


iH  CO  t-  CO  CO 
T-  X  1-0  CI  X 

CI     —     1—    T-l     O 

C1  CI  CI  CI  c^ 

C  C   C:   O  O 


m  o  lo  ir3  lA 

O    ?D    t-    00    OS 
(M    -M    (M    -N    (M 


+ 


O         f-H    (N    CO    -^ 

CO  CO    CO    CO    CO 


lO 

if5  »c^ 

to 

»n 

o  >o 

lO 

ira 

in 

•n 

lO 

>re 

ira 

to 

1-    00 

Ol 

o 

-H    CJ 

CO 

^ 

lO 

m 

1— 

en 

re 

CO 

00     00 

CC 

■* 

TJH      TJH 

-* 

■* 

M< 

•*• 

■(t 

•* 

'^ 

+ 


^kl' 


CI  I-  CI  t-  CI  l~ 

I*  X  CI  -*  CO  CO 

'O  CI  C  t- f  T-i 

CO  CO  :o  CI  CI  ci 


c<i  t-  c^l  t-  CI 

CO  -t<  CI  X  '-h 

X  lO   CI  X  UO 


t-  cq  t-  c^  t^ 

X  CI  -*  CO  CO 

■^  X  -f  O  CO 

C'  C-.  c;  cr.  X 

th  O  O  O  O 


CI  t-  e<)  t-  cj 

CO  I:)"  Cl  X  -* 

CI  X  -f  C^  1-0 

X  I-  I-  CO  CO 

C'  C'  o  o  o 


t-  C<)  t-  C<1  t- 

X  C)  -*  CO  CO 

o  CO  —  o  th 

CO  1-0  lO  -f  'I' 

O  O'  o  c  o 


Interval 


1-0  CO  t-  X  o;  o 

CI  CI  CI  C<l  CJ  CO 


T-(  CI  00  -*  i-O 

CO  CO  CO'  CO  CO 


CO  t-  X  C3  c 
CO  CO  CO  CO  -* 


•^  ci  cr  -rf 


CO  t-  X  05  o 


c 

(a 

'S 

(p 
o 
o 

a) 


-IS 
+ 

'sis 


O    CO    lO    CO 
rj<     Tt<     "^     T^ 


CO  C5  X  C<)  CO  C-1 
CO  CO  C^l  CI  iH  o 

CO  CO  CO  CO  CO  CO 
CO  CO  CO  CO  CO  CO 

C'  o  o  o  o  o 


X  CI  --)<  CI  C5 
X  l^  lO  CO  o 
CI  CI  C<1  CI  CI 

CO  CO  CO  CO  CO 

O  O'  o  o  o 


C0-*C0O-)<  cococoxo 

X  1-0  CI  —   O  1-1  I-  CO  X  -* 

7—   ■— IrHOO  CO~XX 

CO  CO  CO  CO  CO  CO  CI  CI  CI  CI 

O  O  C'  o  o  o  o  o  o  o 


O  X  Tjt  t-  X 
05  CO  X  CI  CO 
t~  t^  CD  O  lO 
CI  CI  CI  CI  CI 

C  O  C'  o  o 


+ 


+ 


+ 


+ 


+ 


%\- 


(M    O    O    OS 


t-    CO    -rp    CO       ^ 


CO    GO    CC    t^       t-         t—    t^    £^    t— 


O    00    CO    CO 


O  CO  t^  O  CI  lO 

O  X  CO  l-O  CO  1-1 

C  O  i-(  CI  CO  -f 

a  o  c  o  o  o 

O  O  C  O  CO  o 


CO  X  X  X  t- 
O  t-  lO  CO  tH 

'f  lo  CO  t^  X 

o  o  o  o  o 
O'  o  o  o  o 

..... 


'^  tH  t-  ,H  -"i* 
O  t^  '^  C?  05 

X  C5  O  -H  iH 

O    O   -r^    1-^    tH 

o  o  o  c  c 

..... 


lO  lO  CO  03  CO 

CO  CO  O  CO  CO 
CI  CO  "^  — f  lO 


CO  CO  T)<  O  CO 

crs  ic  ^H  L-^  CI 

lO  CO  (-  I-  X 


SOOCCO        ooooo 


+ 


m       if2   W3   ic  i:^ 

O  1-"     CN     CO    -^ 

+ 


l^  CI  I-  CI  I-  CI 

CO  CO  -*   M  X  -(< 
CO  CO  CO  CO  1-0  o 

CO  CO  CO  CO  CO  CO 


I-  CI  t-  CI  t- 

X  CI  •*  CO  CO 

-l<   -)<  CO  CI  -rt 

O  CO  CO  CO  o 


CI  t^  CI  t^  CI 

CO  -t  CI  X  -* 

O  05  X  CO  10 

CO  1-0  10  uo  m 


t-  CI  t-  CI  t- 

X  CI  -f  CO  CO 
CO  CI  O  X  CO 
l-O  lO  lO  -*  -* 


CI  I-  CI  t^  CI 

CO  -*  CI  X  -* 

n<  CI  o  I-  lo 

-+  -^  -*■  CO  CO 


I 


liilorval 


O  tH  CI  CO  -*  lO 

c  ©  o  o  o  o 


CO  b-  X  Oi  O 

O  ©   O  ©  ■rH 


I  CI  CO  -f  lO 


CO  t-  X  05  © 

r-l   r^   ^H   1— (  CI 


i-*  C-1  CO  Tf  LO 
CI  CI  CJ  CI  CI 


Table  V.  —  Stirling's  Coefficients  for  F\T). 


227 


K 
O 
li, 

tc 
H 
5^ 
Id 

a 
o 
u 

> 
1 

5 

I—    O    CD    CO    >0       -P 

^    d    C-l    ^        OS 

—      -^      -<      -H           O 

2'=g3    a 

rH  c;   CO   »o     -r 
c;    =-.  o   C-.      S 

-H     Ol    t-     lO 

cr.  X  CO  CC 

1 

1 

+ 

1 

"sis 

O  t^  CO  C5  o  © 

OO  C2  tH  CI   '^   CO 
CO   -*   O   t^  CO  C3 
Cq  CI  CI  CI  CI  CI 

O  O  O  'O  s  o 

1 

■*  CO  o  en  CO 

t^  CO  O  1-1  Cl 
O  -r^  CO  '*  la 
CO  CO  CO  CO  CO 
O  C'  o  o  c 

1 

C-l  1-i  OO  ^  CO 
CO  -*  -*  10  i.O 

CO  l~  OC  CJ  'O 

CO  CO  CO'  CO  -t 

c  s  o  o  o 

1 

rH  Cl    rH    35  r:)< 
CO  ©  ©  lO  lO 
1-H   Cl   CO  -f  1-0 

-*  -f  -*<-)>  -t< 

©©©©'© 

1 

CC  Cl  CO  IC  © 
-t  CO  Cl  ^  © 
©  l-  'X  c  © 

-f   -f    -T   -f   LO 

©  ©  ©  ©  © 

..... 

> 

Id 

5 

(N    3i    t-    O    M       O 
O    C    -^    0-1    CO       -1< 
(M    c-1    I?!    CN    fM       fN 

+ 

Ci    t-    ift    rt<       (M 

^  »c  CD  1—     a: 

2^    tM    c-1    C^       (M 

rH     O     OO     t^         CD 
C5    O    O    rH       (M 

c-1    CO    CO    CO       00 

»0    1(3    tP    CO       CO 
00     Tt^     lO     O         t- 

O?     CO    CO     CO        CO 

CT    (M    (M    ^ 

CO    CO    ■^    -f 

+ 

a  12 
1 

iH   CO   CI   C5  '^   t- 
OO  CC   C3    O   CO  CO 
t-  C-.  i-(  ';)<  CO  CO 

O   O    tH    T-H    ^H   1— ( 

q  o  q  'O  o  o 
+ 

t-  CO  CO  00  en 

O  110  1-1  t-  lO 

•^  CO  CO  CO  1-1 
Cl  M  Cl  en  CO 

q  c  o  o  o 

+  '  '  '  ' 

-*  iO  lO  CO  o 

CO  c-1  Cl  CO  10 
r:t<   t~  C   CO  CO 

00    CO    rt    rt    r-Ji 

O  c:  c  o  o 

+  '  '  '  ' 

+  .04976 
.05311 
.05656 
.06010 
.06373 

+  .06746 
.07128 
.07520 
.07922 

+  .08333 

T 

i5 

»r5    lO    »ft    »0    tra      lO 

O    O    I—    OO    Ci       o 

i^  t-  t-  t-  ir-     GO 

+ 

O   »0    lO    lO       o 
^    IM    CO    -*       O 
00    CC    03    CO       '-/3 

lo  o  lo  in     o 

CO    t-    CO    Ci       o 
X    CO    CO    00       o 

m  i.o  if3  ip     ic 
-H  c-1  CO  ^     in 

O    C3    O    C2       o 

iO    lO    >0    >0 

+ 

-to 
1 

00  CO  yj  CO  00  CO 

lO   -rH    t-    W    CO   CO 

-*  CI  C5  t~  lo  CO 

^  CI  Cl  CO  -*  1.0 

+ 

CO  CO  GO  CO  CO 
CO  "O  l^  tH  o 

1-1   C5   t^  CO   -* 

CO  10  t^  CO  o> 

^^    lH    1-4    1—1    1—1 

+  '  '  '  ' 

CO  CO  CO  CO  CO 

rH    t-   lO   CO   CO 
CO  rH  O   C5  CO 
O   rH  Cl  ei   CO 

en  Cl  Cl  en  en 

+  '  '  '  ' 

X  CO  03  CO  00 
CO  10    t-  rH   1.0 
t-  ©  LO  i-O  -*< 
-*  1.0  ©  I-  X 

en  Cl  ei  ci  ei 

+  '  ■  '  ' 

+  .29413 
.30378 
.31353 
.32338 

+  .33333 

Interval 

s 

lO  CD  t~  00  C3  O 
I-  L-  t-  t~  I-  00 

r-^  Cl   CO   -^    l-O 

CO  CO  CO  CO  <» 

CO  ,1^  C«'   C2   O 

CO  CO  CO  00  C5 

TH  en  CO  -:)<  l-O 

C5  Ci  Cl  Cl  Cl 

©  1^  X  c.  © 

'©  ©  ©  ©  © 

o 

r^ 

o 

a; 
H 
15 

O 

b 
Ii. 
W 
O 
Q 

> 

to 

5 

L^    -^    r;;    00    o       o 

^    Ol    OC    M*       -^ 

i.o   CD   CO   CO      r- 

t-    CC    t-    X       CO 

X     X    1-    X 

1                                                                  1 

^  Ice 

+ 

'l 

1:1^ 

era  -i<  00  1-I  CO  t^ 
CO  CO  >-0  l-O'  -*  CO 

tf    CO   CI   -rH   O   O 

O  O  'O  o  o  o 

O'  O  'O  O  C5'  O 

t-  00  O  CO  i-^ 
t-  a;  o  -^  Cl 

^  Cl  -*  lO  CO 

O  C'  o  o  o 

O  'O  'C:  'O  O 

iH  CO  en  CO  ^ 

-*  1-0  t-  OC  o 

t^  00  C5  O   Cl 
O  'O   O  rH  rH 

o  o  o  o  o 

rH   X    ©  00-rH 

en  CO  o  t-  Cl 

CO'  -*   LO   ©  1- 

y-\   -^i  T^   y-i  T^ 

O  '©  o  ©  © 

o  t-  lO  en  © 

©    Cl    rt   CC    X 

©  ©  1-1  Cl  CO 
rH  Cl  ei  en  ci 

©©'©©© 

+         +  1      1                1                1                II 

> 

-*   OS    »0   Cl   ?o      o 
Tf   ■r:^    O   o   CD      I:- 

+ 

t-  C<I  t-  --*      o 

1:-    CC    00   0>       O 

lO    c-1    C3i    ^       (M 
O     rH     r-     (N         CO 

t-     IC     rH     00        CD 

CO    ^    »0    O       CD 

en  oi  t-  It* 

t-    t-    X    OS 

+ 

=  12 
1 

CO  C5  O  LO  CO  O 
OO  00    CS  CO   t^  iH 
O  'O'  C5  Cl  CO  00 
Cl  CI  1-1  1-1  ^  1-1 

o  c:;  c  'C  c  o 

O  CO  iH  ^  O 

-*  CO   OO   C!   O 

t-  CO  1.0  -t<  ^ 

T— t   1— <  rH  1— ^   1— ( 

C  O  'C  'O  o 

O  l-O  CO  ■*  C' 
O  C2  CO  CO  -t 

CO  1-1  O  Ci  CO 

rH    -rH    rH    O    O 

O  O  CO  o  o 

CO   ^   ©  LO   t^ 
©    1-   Cl    t^   rH 

t-  LO  •*  en  1-^ 
©  ©  c  ©  © 

©  ©  ©  ©  © 

©  rH  ©   t-  rH 

-*  en  ©  X  CO 
©  en  It  LO  t- 

©  ©  c  ©  © 

©   ©  C  ©  '© 

1                   1                1                1           1    +         + 

T 

la 
5 

lO    iC     irt     »ra     1^        lO 
O     ^     CM    O'J    -t        »0 
O    »C    O    i--^    tr?       ^^ 

+ 

O   if5    o    >o      o 
CD    1-    00    C5       O 
O    O    O    O       CD 

O    O    1-0    IC       1/0 

rH    c-1    CO    rtl        lO 

CD    CO    O    CD       CD 

\0    O    lO    ICO       iO 

CO    t^    CO    Cl       o 
CO    CO    CD    CO       t- 

lo    "A    >f2    IC 
1-H     (M     CO     ^ 

t-  r-  t-  I:- 

+ 

-to 
1 

t-  Cl  t-  CI  t-  Cl 

CO  CO  ^  Cl  CO  -* 
tH    CO   rH   CO  O  1.0 
^  CO  CO  Cl  C>  1-1 

O  O  C'  C  C  'O 

t-  Cl  CO  CO  CO 
as  Cl  1-0'  CO  CO 

C5   ^   iH   t-  CO 
C   O  O   O  rH 

00  CO  00  CO  oo 
CO  LO  t^  rH  lO 
Ci  lO  rH  £0  "^ 

rH  en  CO  CO  '^ 
o  c  ©  o  o 

CO  so  CO  CO  00 
1-1   t-  LO  00  CO 
1-1    t-    ^    rH    00 

LO  lO  ©  t--  t- 

©  ©  ©  ©  © 

X  CO  X  CO  X 
CO   LO    t^  1-H   LO 
LO  Cl  ©  t--  -* 

X    ©    ©    ©    tH 
©  '©    ©  rH   rH 

1 

1  1  + 

+ 

+ 

+                           + 

Interval 

S 

O  iH  Cl  CO  ^  »n 
lO  lO  lO  1-0  lO  lO 

CO  C-  00  C5  o 
lO  O  iC  lO  CO 

-^  en  00  -f  lo 

CO  CO   CO   CO   CO 

©  t-  x  ©  © 

©  ©  ©  ©  I- 

rH    Cl    CO    -f    10 
t-   I-   t^    t,    t- 

o 

"-^ 

to 


o 
o 

S 


o 


228 


Table  VI.  —  Bes.sel's  Coefficients  fok  F'{T). 


■f. 
V, 

3 

b 

'A 
O 

O 

> 

fa' 

Cl     1--     -O     Kt.     -^        -rt* 
^1     Tl     M     G^l     ?J        <N 

+ 

CT     ^    Cl    O       CO 

CO    O    -rt    CO       -^ 

T-*     C5     00    I-        CO 

^    CO    Cl    1-1 

+ 

-1 

+ 

't  CO  o  ;r;  ^  o 

CV  M  1-0   I-  O  CI 

O   ■r^   '-1   r-(  CI    CI 

o  o  o  c  =>  o 
c  o  o  o  o  o 

+ 

+  .00249 
.00271 
.00292 
.00311 
.00330 

00  '*  o  •*  t- 
•>*•  CD  CO  OO  o 
CO  CO  CO  CO  -t< 

o  c:>  o  o  o 
o  o  o  o  o 

+  '  '  ■  ' 

CO  C5  CO  CD  CO 

i-H  Cl  eo  -^  LO 

^  -*  -*  -t  •* 
o  o  o  o  o 
o  o  o  o  o 

+  '  '  '  ' 

05  eo  CD  t/o  05 

LO  CD  CD  CD  CD 
-*   -*  -f   "Cf   ^ 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

+  '  ■  ■  + 

> 

fa' 

i5 

CO     «    CO     K-?     X        CS 
r-    X    CC    X    CO       X* 

7  '^""" 

T-    CO    »«    CD       CO 
OS    C5   O    05      c: 

Oi    ^    Cl    fM       ^ 
CS    O    O    O       CO 
^     Cl    O     !M        Cl 

LO   lo  ^-  CO     CO 

o  cc  o  o     o 

Cl    Cl    Cl    Cl       Cl 

1-   X    O:    X 
(N     CN    (N     C^ 

1 

1 

00  O  O  O  tH  CO 

't  t-  cc  c  CI  CO 
C:  l^  '0  -f  CI  O 

-+     -t-     -f     ^     Tf     -1< 

o  o  o  o  o  o 

-*  CO  O  13  05 
■*  lO  CO  CO  CO 

OO  CO  -t"  Cl  o 

CO  CO  CO  CO  CO 

o  o  o  o  o 

T-i  Cl  tH   05  t- 
t-  l^  t-  CD  CO 
CO  CO  -l<  Cl  o 
Cl  Cl  Cl  Cl  Cl 

o  c  o  o  o 

CO  OO  eo  CO  o 

CO  lO  lO  -^-  -*< 
CO  CO  "-f  Cl  o 

T-^     T^     T^    T^     T^ 

O  CD  o  o  o 

Cl  LO  I-  00  © 

CO  Cl  r^  ©  © 
<Z>  CD  1*  Cl  © 

©  ©  ©  ©  © 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+       + 

n 

fa 
(5 

O    to    O    JfS    »o       »o 
^     CO     C^     ^     O        Oi 
(M     CJ     CN     !M     3^        T-. 

1 

lO     lO    lO    o       »^ 

00    t-    CO    O        rP 

»0    lO    ic^    »o       to 

CO    Cl    ^    O        OS 

lo  »ro  >o  to     to 

00    t-    CO    to       '* 

to  to  to  to 
eo  Cl  1-1 

"12 
+ 

1 

0>1  t^  CI  t^  C1  t- 

-f  ->:  CI  T  -o  o 

C   Cl  1.0   I-  c3^  <-^ 

— 1   r^   ^   T--   T-l   CI 

o  o  o  c  o  o 

Cl  t-  Cl  t-  Cl 

CD  ^  Cl  OO  -:1< 
CO  O  t-  CO  o 
Cl  Cl  C^l  Cl  CO 

o  o  o  o  o 

t^  Cl  t-  Cl  t- 

00  Cl  -*  CO  CD 

■r-l   CO    -*    O   CO 

CO  CO  CO  CO  CO 

o  o  o  o  o 

Cl  t-  Cl  t-  Cl 

CD  -i<  Cl  OO  TjH 
t^  CO  C--  C-.  o 

CO  eo  eo  CO  -* 
o  o  o  o  c 

t-  Cl  I—  Cl  t- 

£»  Cl   -f    CD    © 
O    i-H    1— '   1— 1   1— t 

^  -ll  -f  •*  -* 
©  ©  ©  ©  © 

1                  1               1               1               1          1 

n 

1 
S 

LO  -^  CO  CO  tH  O 
C)  CI  Cl  Cl  C)  Cl 

d 

1 

C5  00  t-  CD  lO 

•*  CO  Cl  i-(  o 

1H   T-l   tH  tH  T— 1 

Oi  00  t-  CO  lO 

q  CD-  o  q  q 

-1*  eo  Cl  1-1  © 
©  ©  ©  ©  © 

■  ©■  © 

1 

Interval 

^ 

LO  CD  t-  OO  O  O 
Cl  Cl  CI  Cl  00  CO 

1-H  Cl  CO  ■*  lO 
CO  CO  CO  CO  CO 

O  I-  CO  o  o 
CO  CO  CO  CO  ^ 

^th  Cl  CO  ^  lO 

©  t-  00  05  © 
'Cf    Tj<    'Jjl    >*    LO 

o 

© 

o 
w 

H 

5 

b 
O 

> 

1 

5 

— ■     -M     —     71     ^        v- 

^    -^   -IH    -r    -^      -i^ 

+ 

^   ^   C^'   o--^      o 
^   -i^  -r   CO     Ti^ 

GO     CO     00    1^        CO 
CO    CO    CO    o^       CO 

CO    -t    to    CO       o^ 
CD   CO    CO    or      00 

-H      ^      O      CS 

00    CO     00     Cl 

+ 

CO  Cl  o  c;  1-  'O 
CO  Ci  LO  O  -O  Cl 
CO  t-  t-  t-  o  o 

o  o  o  o  o  o 
o  o  o  o  o  o 

00585 
00544 
00504 
004G4 
00425 

O   t-  C5  i-H   -)< 
CO  -*  O  I-  CO 
.  CO  CO  CO  Cl  Cl 

o  o  o  o  o 
o  o  o  o  o 

<»  Cl  CO  00  o 
C3  CD  Cl  C:  CD 

1— 1    ^H   1— i    O    O 

o  o  o  o  o 
o  o  o  o  o 

r^  ^  ic  L5  Tji 

Cl  ©  CO  ©  Cl 
c  ©  ©  ©  © 

©  ©  ©  ©  © 
©  ©  ©  ©  © 

1                   1                1                1                1  +       + 

> 

5 

»r^    .-■    -^    O    «i^       O 

X    Gl    O    O    O       O 

■n<     X     !M    CO       -. 
rH     .-H     fN     G^         CO 

-*    CO    Cl    LO       CS 
CO    CO    ^    -^       •* 

d    to    CS    Cl        to 
to    O    to    CO       CD 

CO    O    CO    CO 

CO  r-  L-  t^ 

1        T     II 

1 
1 

CO  C»  t-  ^rH  iH   CD 

CO  rl*  O  CO  o  lO 

CO   C-1    rH    O    O   OO 

OO  CO  CO  CO  I-  I-- 

c  o  o  o  o  o 

t-  CO  >0  CO  b- 
-*   CO  1-1  C3  CD 
l^  CO  lO  CO  Cl 

t-  I-  t^  t-  t- 

o  o  o  o  o 

CO  Cl  TiH  Cl  t- 

CO  O  CD  Cl  t-- 

i-(  o  a;  t~-  lo 

I-  t-  CD  CD  CO 

o  o  c  o  o 

OO  CO  i-l  Cl  o 
Cl  t-  Cl  CD  O 

^  Cl   ^   C5  OO 

CD  CO  CD  lO  LO 
O  C  CD  O  O 

lO  t-  t-  -*  00 
CO  ©  ©  Cl  ->< 
©  -)i  Cl  1-1  © 
IC  10  LO  lO  -^ 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+         + 

T 

fa' 
5 

1.'^    »f5    lO    iC    >0       lO 

Ci    X    i—    O    lO       "^ 

rji       -^      Tp       -^       Tfl            ^ 

1 

lO    »o    O    i.O       itO 
CO   Cl   ^    o      c» 
■*    rjl    1^1    TJH       CO 

»0    lO    »0    ».C       ICO 

00    t-    CO    Ui       ^ 
CO    CO    CO    CO       00 

to  to  to  to     to 

CO    Cl    T-H    O       <3S 
CO    CO    CO    CO       Cl 

to    to    to    to 
00    t-    CD    to 
W    Cl     d    Cl 

i 

CO  OO  CO  OO  CO  OO 
CO  CO  '-0  t-  tH  >o 

CO  (»  CO  CO  '^  o 

03  b-  t-  cr>  CD  lO 

o  o  o  o  o  o 

eo  00  CO  CO  CO 

tH  t-  «0  00  CO 
lO  O  CO  Cl  OO 
LO  O  '^   '*  CO 

o  o  o  o  o 

00  CO  OO  CO  OO 
CO  LO  t-  T-l  lO 
-*  O  CO  CO  03 

CO  eo  Cl  Cl  th 
o  o  o  o  o 

CO  CO  CO  00  CO 

iH  t-  LO  eo  00 

CD  Cl  C5  CD  CO 

1-1   -rH   O   O   O 

©  o  o  c  o 

00  l^  Cl  l~  CO 

CO  ^  Cl  00  -* 
©  Cl  LO  t^  © 

©   ©   ©    ©   tH 

©  ©  ©  ©  © 

+ 

+ 

+ 

+ 

+  1        1 

n 

Hot 

O  Ci  OO  l^  CO  UO 
1.0  ^  -*  'ri*  -*  tH 

'^  CO  Cl  iH  o 

^^    '•^^    "^    ^T    ^T 

05  00  t~  CD  lO 

CO  CO  CO  CO  eo 

-*  eo  Cl  ^  © 
eo  CO  00  CO  00 

©  CO  t-  ©  lO 
Cl  Cl  Cl  Cl  Cl 

© 

1 

© 

1 

Interval 

S 

O  — 1  Cl  CO  -^^  "O 

c  o  tr  o  o  o 

CO  l-  CO  Ci  o 
C  O  O  O  --1 

iH  Cl  eo  'Ttt  lo 

tH  ^H  1-H   1-H  tH 

CO  t-  CO  as  © 

tH  r-l  1-1  1-1  Cl 

1-1  Cl  eo  't  lO 

Cl  Cl  Cl  Cl  Cl 

o 

© 

Table  YI.  —  Bessel's  Coefficients  for  F'{T). 


229 


O 

h 

a; 

« 
O 

o 

> 

(5 

05     O      rH      rH      O^          CO 

(N    CO    CO    CO    05       CO 
1 

lO    -t    CD    CD       t- 

CO    CO    CO    CO       CO 

Ol    CO    CO    O       OS 
CO    CO    CO    If       CO 

O    O     rH    -H        rt 

-^      Til     It      11*          ■* 

Ol     rH     Ol     rH 

'^  -)<  -t  •* 

1 

r 

+ 

"sis 

-*  »0  lO  '^  t-  o 
C5  'O  CO  O  C-l  CO 

c  o  o  c  o  o 
o  o  c  o  o  o 

o  o  o  o  c  o 

CO  a>  01  00  •* 

C3  Ol  O  C3  CO 
O   r-   rt   -^    M 

O  O  C'  o  o 

c  o  o  o  o 

•pH   C3  1-  1.0  lO 
l^  O  -t  OO  Ol 
Ol  CO  CO  CO  -+ 

o  o  o  o  o 
o  o  o  o  o 

icH  ^  T)<  l.O  CO 
O  O  -t<  !»  Ol 
-*  lO  1-0  1-0  CO 

O  O  O  C'  o 
O  O  CD  o  o 

t-  Cs  O  01  CO 

CO  O  10  C^  CO 

CO  t-  t-  t-  aj 

o  o  o  o  o 
o  o  o  o  o 

+       +  1 

1 

1 

1 

1          1 

> 

la 
5 

O   CO    O    00    >o       M 

t-  t-  I-  CO  as     o 

c:5  ira  (M  Ci     no 

O    K5    lO    -*        •* 

!M     00     Ttl     r-1         03 
'*    CO    CO    00       o^ 

(M  CO  tC  oa     .n 

(M    rt    rt    o       O 

8§SS 

„^„^^^„„„                                                           ill 

-IS 

+ 

s|2 

1 
■^'sl-' 

00  -^  t-  t^  "rt  o 
^  C-1  C^  O  CO  o 

C5  -n  1-1    •*    CO  00 
^  UO  lO  lO  "O  lO 

o  o  o  o  o  o 

01  ^  CO  CO  t- 
CO  ^1  i~  oi  i^ 

C5  ^  01  -f  o 

O  CO  O  CO  CO 

o  o  o  o  o 

Ol  ^  Ol  CO  1- 

Ol  CO  O  CO  CO 
t^  CO  O  1-1  Ol 
CO  O  L-  t-  t- 

o  o  o  o  o 

00  l-O  CO  t-  CO 
C2  tH  CO  -)<  0 

CO  o  CO  t-  CO 
[_  I-  l^  l^  t^ 

o  o  o  o  o 

1-1  ^  t^  OO  CO 
CO   CO  LO  rjl   CO 
C5  C  rt  01  00 
t-  Oj  CO  O)  CO 

o  o  o  o  o 

1                  1               1               1               1          1 

n 

Q 

»f2   »f5  lO  o   ifr     uD 

iC    :0    IT-    CO   Oi       O 
CM     (M     Cl     Ol     (N        CO 

+ 

lO  lO  lO  lo     in 

rH     C-1     CO     "^        >0 
CO    CO    CO    CO       CO 

lO    >0    >C5    iO       o 

CO   r-  CO  o;      o 

CO    CO    CO    CO       '^ 

»r5  >c  >o  »o     lO 

^    Ol    CO    i*       no 

-t  Tl<  -Jl  •*     -* 

m  in  iO  >ft 

CO     t-     X)     C2 
-*    Tl*     rt*     -I- 

SIn 

1 

01  t-  c-i  t-  cc  CO 

-*  CO  Ol  -*  CO  CO 
O  I-  O  C-1  O  CO 

tH  o  o  o  o  o 
o  o  o  o  o  o 

00  CO  00  CO  CO 
CO  lO  t^  1-1  lO 

O  C2  Ol  CO   05 
O  O   iH   -H   tH 

o  o  o  o  o 

CO  wo  CO  OO  CO 
T-l  l^  lO  CO  CO 
CO  CO  O  ^  CO 
01  Ol  00  CO  CO 

o  o  o  o  c 

00  00  CO  CO  CO 
CO  lO  t^  T-l  l-O 

01  CD  O  1-0  C5 
-+  -ti  LO  lO  lO 

c  o  o  o  o 

CO  00   CO   CO  CO 
tH  1--  lO  CO  00 
-t  OO  00  00  CO 

CO  CO  t--  t-  OO 
o  o  c  o  o 

1        1  + 

+ 

+ 

+ 

+         + 

n 

>0  CO  l-~  CO  C3  o 
CI  C^l  C-I  0)  01  CO 

o 

+ 

■rH    01    CO   '^    O 

CO  CO  CO  CO  CO 

CO  t-  a;  0-.  o 

CO  C0_  0^  00  Tt* 

^  Ol  00  ^  lO 

•^  -*  ^  ■*  -* 

CO  1^  CO  C5  O 
^  T)<  -I*  ^  lO 

■    ■    ■    ■  o 

+ 

Interval 

s 

lO  CO  t-  CO  OS  o 
t-  I-  t-  I-  t-  00 

iH  Ol  CO  -f  lO 
CO  CO  OO  CO  CO 

CO  t-  CO  C5  o 

OO  OC  OO  CO  05 

1-1  Ol  CO  -f  lO 
'^  1^  Gi  Oi  Oi 

CO  t--  00  C5  o 
O  CS  Ci  Oi  o 

o 

lH 

o 
b. 

H 
Z 

5 

b 
fa 
H 
O 

o 

> 
-1 

In 
5 

rt     «    D5    -*    CO       t- 

1 

CO    Ol    —    — 1       05 

•+   CD  CO   O)     c; 

O    1—    01    -il       -t* 

lO    CO    1-    0-. 
C-I    Ol    CI    Ol 

1 

-A 

1 

'sis 

O  CO  CO  CO  O  CO 
CO  CO  CO  O  lO  lO 

^     ^     -*     'I*     -*     -Tfl 

o  o  o  o  o  o 
o  o  o  o  o  o 

O  CO  O  OO  t- 
-*  00  01  T-<  o 
^  -*  -*  -f<  -H 

o  o  o  o  o 
o  o  o  o  o 

-*  O  -f  00  o 
C:  OO  CO  •*  00 
CO  CO  CO  CO  CO 

o  o  o  o  o 
o  o  o  o  o 

1-1  01  1-1  O  lO 
1-1  C2  t-  -*  Ol 
CO  Ol  Ol  Ol  Ol 

o  o  o  o  o 
o  o  o  o  o 

iH  CO  O  00  T* 

0  t-  LO  Ol  o 

01  1-1   iH  -rH  O 

o  o  o  o  o 
o  o  o  o  o 

+ 

+ 

+ 

+ 

+         + 

> 

fa 

5 

CO    C»    l»    t-    O)       CO 

o  o  o  o  o     o 

0^    O    (M    tM    (N        in 

^-   »o   lo  1*      <N 

o  o  o  o     o 

0-1    C-1    Ol    <M       C-) 

<M    ^    CS    00       CD 

O    O    <31    C»       CS 

IM       (M       T—       ^            T— 

lO    00    ^    05       CO 

a:  Ci  05  CO     CO 

O    CO     •-'     CO 

00  00   CO  r- 

1— 1      rH      i-H      rH 

1 

-12 

r 

1 

O  CO  1--  lO  Ol  o 
O  O  tH  01  CO  -f< 
C  Ol  -*  CO  00  o 
o  o  o  o  o  ^ 
o  o  o  o  o  o 

CO  CO  00  CO  t- 
^  lO  l.O  CO  CO 
C-1  -l<  CO  CO  o 

^H    ^H    T— 1   -rM    01 

o  o  o  o  o 

05  1-1  0-1  tH  Oi 
CO  t-  I-  t^  CD 
Ol  -H  CD  CO  O 
Ol  01  01  Ol  00 

o  o  o  o  o 

LO  O  CO  -cf  OO 
CO  CO  lO  "*  00 
Ol   -rt*   CO  CO  O 
CO  CO  00  CO  -* 

o  o  o  o  o 

iH  CO  05  O  00 
Ol    O    OC    I-    rjH 
Ol   -*  LO  t--  C5 

-*  ^  ^  -)<  -*( 

O  O  O  C'  o 

1 

1 

1 

1 

1           1 

n 

fa 

lO     11^     iC     IC     O        lO 
1-1    G^    CO    '^       O 

+ 

>0   O   lO   to      >ra 

CO  1-  00  c:i     o 

»0    lO    O    lO        lis 

w    Ol    CO    '^        tf5 

lO    »0    lO    4iO       no 

CD  r-  CO  o     o 

rH      — t      T-H      T-l          Ol 

lO   o   »o   to 

.-H     CM     CO    Ml 
CN    CI    iM    C^ 

+ 

-12 
+ 

1 

t^  Ol  t-  Ol  t^  01 

CO  CO  -+  01  00  -^ 

^H     T-(     T^    1— 1    O     O 

^  -*  -H  -t<  -+  -t< 

O  O  O  C'  o  o 

t-  Ol  l^  Ol  t- 

CO  01  ^  O  CO 
C5  O  OC'   I-  CO 
CO  CO  CO  CO   CO 

o  o  o  o  o 

Ol  I-  Ol  t^-  01 

CO  -*  01  CO  -c)< 
O  -)<  CO  i-^  o 
CO  CO  CO  CO  00 

o  s  o  o  o 

t-  01  t-  Ol  t- 

OO  Ol  -!f  CO  CO 

00  l—  LO  CO  1-1 

01  01  Ol  Ol  Ol 

o  o  o  c  o 

Ol  I-  01  t-  01 

CD  -t  Ol  CO  -* 
0>  t-  IC  Ol  o 

lH  iH  iH  1-*  iH 

O  CD'  o  o  o 

1                   1                1                1                1           1 

n 

1 

C  — t  Ol  CO  'f  lO 

o  o  o  o  o  o 

CO   t-  CO  C5  O 
O  O  O  O  1-1 

■r-l  Ol  CO  'cf  »C 

■r-t     T~>     T-t     ^-i     T^ 

to  b-  OO  O!  O 

1-1  iH  1— (  1— (  Ol 

iH  0-1  CO  "*  lO 
Ol  Ol  Ol  Ol  01 

o 

+ 

o 

+ 

Interval 

s 

O  1-1  IM  CO  -f  lO 
lO  i-0  »0  lO  O  1.0 

CO   t—  (»  05  O 
1-0  l-O  UO  O  CO 

t— 1  Ol  CO  "^  lO 

CO  CO  CO  -O  CO 

CO  t^  OO  CS  o 

CO   CO  CO  CO   t- 

^  Ol  CO  1*  LO 
t^  b-  t-  t-  I- 

o 

o 

DR.  Gc 


230 


Table  VII. 


0) 

■73 


s 

o  "-J  CI  ^■:  -t  >-•: 

o  s  c  =  o  o 

O  O  C:   C:   C:   O 

OOf) 
007 
008 
001) 
010 

rH    CI    CO    -»■    10 

c  ©  S  ©  © 

010 
017 
018 
019 
020 

021 
022 

02;; 

024 
025 

O 

© 

o 

K 

b 

O 

s 

■< 

O 

d 

c  o  c  '^  CI  ^•5 

-*  1-   ©  X  © 

rH 

CI  rj-  t-  ©  CO 

rH  T-l  rH  CI  CI 

©  ©  CI  ©  © 

Cl  Cl   CO  CO  rf 

't  OO  CO  GO  CO 
rt<  TJH  lO  l-O  © 

d 

©  O  O  •rH  C1  CI 

«  »0  to  00  © 
rH 

rH  -*  ©  ©  T^ 
TH  rH  rH  rH  CI 

rl<   t^  rH  ^   OO 
CI  CI  CO  CO  CO 

CI  ©  ©  lO  © 
^  -*  >0  lO  lO 

r-t 

CO 

O  O  O  T-l  rH  CI 

CO  T(<   ©  t-  ©1 

rH  CO  LO  OO  © 
rH  rH  rH   rH  CI 

CO  ©  ©  CI   © 
CI  CI  CJ  CO  CO 

©    rJ4    CO    CI    © 
■*    -*<    rf    lO    1.0 

00 

r- 

O  O  O  tH  tH  CI 

CO   rf   lO    t-   © 

O   CI   rf   t-  C5 
T^   r^  r-i  tH  T-i 

CI    lO  00   rH   rj< 

CI  CI  CI  CO  CO 

I-  rH  lO  ©  CO 
CO  TJH  r).  rt  lO 

t* 

^ 

O  O  O  tH  1-1  CI 

CO  -*  lO  CC  CO 

©  CI  -H  ©  GO 

©   Ct   ©   ©  CI 
CI  CI  CI  CI  CO 

1.0  ©  CI  ©  © 
CO  CO  "*  Tl*  lO 

CO 
1-1 

lO 

O   O   O  tH   i-H   CI 

CO  rf  J-O'  ©  X 

©   rH   CO   lO   t— 
j-<  r-'  r~i  T~^ 

©  CI  -*  t-  © 
r-  d  C^l  CI  CO 

CO  ©  ©  CO  I- 
CO  CO  ■*  -*  •* 

1.-5 

rH 

-* 

O  O  O  i-H  rH  CI 

CO  CO  -f  ©  I- 

CO  ©  CI  -*  © 

T-t     J-<     T-i     T-i 

CO  ©  CO  lO  CO 
rH  CI  CI  CI  CI 

rH  -^   l^  ©  .^ 

CO  CO  CO  -^  ■* 

Tf 

CO 

C:  O  O  tH  ^  C1 

CI  CO  Tjt  1.0  t- 

00  ©  rH   CO   10 
rH  T-1   rH 

t^   ©   rH    CO    © 
rH  rH  CI   CI   CI 

©  rH   rj*   t^  rH 
CI  CO  CO  CO  ■* 

CO 

1-* 

<M 

O  O  C:  -^  1-1  CI 

CI  CO  -»(  i-O  © 

t-   ©  ©   CI   -f 

rH   rH   rH 

O  l^  ©  C-1  -* 
rH  rH  iH  CI  CI 

©  ©  CI   lO  00 
C<1  d  CO  CO  CO 

IM 

rH 

»H 

O  O  O  O  T-l  tH 

CI  CO  -+  -f  o 

t^  GO  ©  rH  CI 
rH   rH 

rJH  ©  GO  ©  CI 
rH  rH  rH  CI  CI 

rf    t-    ©    CI    -* 

CI  CI  CI  CO  CO 

^ 

o 

O  O  ©  O  tH  T-1 

CI  CI  CO  •*  o 

©  I-  CO  ©  rH 

CO  r)(  ©  X  © 
rH  rH  rH  rH  CI 

C-1  -H  ©  ©  rH 
CI  CI  CI  CI  CO 

o 

§ 

©    ©    ©    ©    r-1    T-< 

CI  CI  CO  -)<  -f 

10  ©  CO  ©  © 
rH 

CI  CO  in  ©  00 

rH  rH  rH  rH  rH 

©  c-l  r*<  ©  00 
CI  CI  CI  CI  CI 

°. 

o 

©  ©  ©  ©  rH  iH 

rH  CI  CO  CO  rjt 

lO  ©  t-  00  © 

©  CI  CO  -1<  © 

OO    ©   rH   CO   l-O 
rH  rH  CJ  Cq  C^ 

CO 

q 

o 

©    ©    ©     C     tH     T-l 

rH  CI  CI  CO  CO 

"*  lO  ©  t^  CO 

©   ©  rH    CO  rH 
r^  r-i  i-<  T-i 

lO  t~  ©  ©  CI 

rH  rH  rH  CI   CI 

q 

q 

©  ©  ©  ©  ©  ^ 

rH   rH  CI   CI   CO 

■*   ■*   UO  ©  t- 

CO   ©   ©    rH  CI 
y~i   T~i  T^ 

CO  lO  ©  t-  © 
rH  rH   rH   rH  rH 

o 
q 

©   ©  ©  ©  ©   •rH 

rH   rH    CI  CI   CI 

CO  ^  ■>*  K5  © 

©  t-  GO  ©  © 
rH 

rH  C^  CO  r*   © 
rH  rH  rH   rH  rH 

g 

o 

©©©©©© 

rH   rH  rH   CI   CI 

CI  CO  CO  rX  rj< 

lO  ©  ©  t^  00 

©  ©   rH  CI   CO 
y~t    T^    T-{    T^ 

rP 

q 

§ 

©©©©©© 

CI  C-l  CO  CO  CO 

rf  rji  lO  1.0  © 

t-  t~  CO  ©  © 

05 

q 

o 

©©©©©© 

©  ©  rH  -H  rH 

rH  rH  C)  CI  CI 

CO  CO  CO  rjl  rH 

-*  LO    10    ©  © 

9 

q 

©©©©©© 

©  ©    ©   ©    © 

»^    rH   CI   CI    CI 

CI  CI  CO  CO  CO 

q 

o 
o 
d 

o  ©  ©  ©  ©  © 

©  ©  o  ©  © 

©  ©  ©  ©  © 

©   ©    ©    ©   © 

©  ©  ©  ©  © 

§ 

d 

H 

O  rH  CI  CO  rj*  ira 

o  ©  ©  ©  ©  © 
©  ©  c  ©  ©  © 

©■ 

CD  t-  00  C5  © 
©   O  ©  O   rH 
©©   ©   ©   © 

rH  C?  CO  '^  »0 
rH    rH   rH   rH   rH 
CO©©© 

©  t-  00  ©  © 

rH    rH    -H   rH   CI 
©   ©    ©    ©    © 

rH   CI    CO   rH   1.0 

CI  CI  CI  CI  CI 

©  ©  ©  ©  © 
'    ■    ■    ■  © 

Giving  ij:     To  he  used  in  finding  u  avhen  J^„,  ls  given. 

XoTE.  —  The  ([uantity  ;/  has  the  same  sign  as  argument  K. 


Table  VU. 


2'.n 


T-H  -tJ 


%/ 

i;;  o  t-  :^  —  O 
CI  01  CI  CI  CI  :■: 

©  o  c  o  c  o 

T-  c)  CO  -t<  "O 

CO  CO  CO  CO  CO 

c  o  o  o  o 

CO  t-  CO'  CJ  o 
CO  CO  CO  CO  ■* 

o  o  o  c  o 

1-1  CI  CO'  -t<  1-0 

-*  -^  -t  -t  -t 

O'  o  c  o  o 

O  I-  CO  CV  o 
-*  -t  -t  -*  'O 

c  o  o  o  o 

o 

> 

o 

s 
s 

a 

K 
<i 

b. 
C 

'■11 

> 

o 
d 

CO  oo  eo  00  '^  o 

CO  O  t-  I-  00  05 

CO  CI  05  CO  CO 

05  O  O  1-1  c? 
1-1  tH  tH  -M 

O  t-  ■*  CI  o 

CO  CO  ^  lO  CO 
iH  1-1  iH  iH  iH 

CO  CO  1.0  -*  CO 

CO  t^  CO  C3  o 

1— 1  1— 1  T-l  1— 1  C<J 

CI  iH  o  o  o 
T^  CI  CO  -*  IC 
C^  CI  CI  C^l  01 

o 
d 

o 

O  -*  Ci  -*  o  o 
lO  to  'i>  t-  CO  OO 

^  t--  CO  O  CO 

O  Ci  O  T^  T-t 

^H  1— <  tH 

CO  C  t-  1*  CI 

CI  CO  CO  Tt<  lO 
■r^  1— (  iH  iH  1-1 

O  00  CO  -♦1  CI 

CO  CO  1--  CO  o 

iH  O  03  00  00 
C  1-1  —1  CI  CO 
CI  CI  CI  CI  CI 

Ol 

00 

CO  tH  CD  tH  CC  -^ 
O  CO  CO  t-  t-  CO 

CO  C1  CO  -1-  o 
CO  O  C5  O  T-l 

T-l  1— 1 

t-  CO  O  t-  1* 

1-1  CI  CO  CO  ^ 
■rH  1— 1  1— 1  1— 1  iH 

•rH  C3  CO  -*  C-1 

o  lo  CO  I-  CO 

tH  T-l  iH  1-1  iH 

O  C5  t^  CO  lO 
C-.  O  O  1-1  O) 
tH  -H  CI  CI  CI 

CO 
1-f 

i-H 

CO  t-  N  t-  T-(  CO 
lO  lO  CO  CO  I-  r- 

C)  1--  00  O)  -* 
CO  CO  02  C5  o 

O  CO  CO  Ci  O 
iH  T^  CJ  CI  CO 
1— (  1— (  tH  rH  1— 1 

CO  O  I-  LO  C<1 

1*  IS  la  CO  t- 

IH  T-l  1-1  1-1  T-l 

O  00  CO  ^  CI 

CO  CO  c;  o  T-l 

tH  1-1  tH  C^l  C-1 

t- 

o 

O  '^  CO  CO  t~  C^l 
O  lO  >0  CO  CO  t- 

t-  CI  t-  CI  CO 

t-  00  CO  crs  C5 

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O  I-H  1-1  CI  CI 
iH  1— 1  iH  1— 1  iH 

-*  th  CO  lo  CI 

CO  '^  -*  1-0  CO 

T— 1  T— 1  1— 1  1— 1  tH 

03  t f  CI  o 

CO  i^  a>  cc  o 

1-1  1-1  T-l  1-1  CI 

CO 

lo 

t-  T-i  >n  05  CO  00 

TJH  lO  lO  lO  CO  CO 

C-1  t~  CI  t^  CI 

t-  t-  00  CO  05 

t-  CO  CO  ■*  o 

05  o  o  ■^  cq 

iH  -^^  ^H  iH 

CO  C-l  C:  lO  CI 

Ca  CO  CO  -rti  lO 

T-l  T— 1  tH  T— 1  1— 1 

C3  CO  CO  O  CO 
LO  CO  t-  CO  CO 
▼H  iH  rH  1— 1  T-l 

lO 

■^  t~  -rH  lO  05  CO 

rt(  Tt  lO  lO  lO  CO 

l-  CI  CO  T-l  CO 
CO  t-  t-  00  00 

1-1  CO  ^  CO  CI 

C3  05  O  O  tH 

^  T-l  1-1 

CO  CO  Ol  CO  CI 
•rH  CI  CI  CO  '^ 
1— 1  1— 1  iH  1— 1  iH 

CO  lO  T-H  OO  la 

1*  lO  CO  -.o  t^ 

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« 

•t^  ^  t-  1-1  lO  CO 
-^  t1<  Tj<  lO  "O  »o 

CI  t^  1-1  iO  o 
CO  CO  t^  t~  oo 

1*  C3  -T)l  03  -* 

CO  CO  03  OS  o 
1—1 

C5  O  O  CO  CI 
O  iH  CI  CI  00 
1— 1  tH  tH  T— 1  T— 1 

CO  -f  O  CO  CI 
CO  -*•  O  l-O  CO 
^^  tH  ^^  T-l  1— 1 

Vi 

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CO  iH  -#  t-  O  -* 
CO  -*  -*  'it  O  "O 

CO  1-1  »Oi  05  '^ 

O  CO  O  CO  t^ 

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1-  CO  CO  03  05 

tH  CO  tH  CO  C<1 

O  O  -^  TH  CI 
tH  1— 1  tH  T-l  T-l 

t-  CO  00  -^  o 
CI  CO  CO  -*  lO 

tH  tH  T-l  iH  tH 

<M 

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CO  CO  ■*  Tj<  Tl<  lO 

CO  CO  o  -rji  ^- 
l-o  lo  CO  CO  CO 

1-1  1-0  C3  -f  CO 

t-  i^  i^  (»  CO 

CI  t-  CI  CO  ^ 

C5  C5  'O  'O  iH 
1-^  -tH  iH 

CO  1-H  t^  CI  CO 

1-H  CI  CI  CO  oo 

tH  tH  ^H  -H  tH 

TH 

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CO  CO  CO  CO  '*  'i* 

03  -^  -f  CO  tH 
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lO  CO  CI  o  o 
CO  CO  t-  t-  00 

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CO  00  o  o  o 
1—1 

CO  O  LO  O  lO 
O  tH  tH  CI  CI 

1-1  1— 1  T— 1  -H  1— 1 

o 

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00  O  CO  lO  CO  T-l 

01  CO  CO  CO  CO  ^ 

CO  CO  05  ca  lo 

Tt  ^  Tj<  la  lo 

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lo  c;  ^  a>  CI 

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lO  t-  05  i-l  -*  CO 
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CO  1-1  -*  CO  S3 
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1-0  CO  T-l  CO  CO 
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lO  lO  lO  lO  CO 

CO  CO  C3  cq  lO 
CO  CO  CO  I—  t- 

CO 

q 

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CO  t~  CO  O  iH  CO 
tH  T-l  1-1  C<1  !M  CI 

■*  CO  t-  c;  iH 
CI  CI  CI  CI  CO 

C-l  -*  CO  CO  o 
CO  CO  CO  CO  -* 

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CO  lO  OO  O  CI 
lO  l-O  O  CO  CO 

q 

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CO  •*  lO  CO  b-  CO 
iH  1— 1  tH  iH  T— t  tH 

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CO  t~  03  O  0^1 
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Tf  ^  1*  ^  lO 

o 

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C5  O  T-(  CI  CO  -i* 

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C3  r-H  CI  CO  -* 
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lO  O  00  03  o 
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CI  CO  LO  CO  CO 
CO  CO  00  CO  CO 

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q 

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CO  t~  t^  CO  CO  C5 

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CO  '#  ^  la  CO 

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CI 

q 

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CO  CO  -*-*■*  IC 

10  lO  1-0  CO  CO 

CO  I-  t-  CO  CO 

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q 

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o  c  o  o  o 

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o 

d 

•-. 

lO  CO  I-  (/)  CJ  o 
CI  CI  CI  CI  CI  CO 

o  o  o  o  o  o 
o 

1-1  C5  CO  ■*  lO 

CO  CO  CO  CO  CO 

o  o  o  o  o 

CO  1^  CO  05  o 
CO  CO  CO  CO  '* 

o  o  o  o  o 

1-1  CI  CO  -f  lO 
1*  -cH  -*  -*  -* 

o  o  o  o  o 

CD  t:-  00  05  O 
-*<-*-*  ^  lO 

O'  o  o  o  o 
■  •  ■  ■  o 

■A 


3 


05 

-c5 


Giving  y:     To  be  used  in 

Note.  —  The  quantity  //  has 


FINDING    n   WHEN   F,,    IS    Gn^EN. 

the  same  sisn  as  argument  A'. 


232 


Table  YIIT. 


Coefficients    fok    Computing 

F„  =  t\  -^  MO,  {F^  +  -:  a  +  B^,  +  Ty). 


n 

BH^ 

Dlff. 

-^m-^) 

Dlff. 

0.00 

0.0000 

0.0000 

.02 

+   .0001 

+   1 

-   .0017 

—17 

.04 

.0003 

2 

.0033 

l(i 

.06 

.0006 

3 

.0050 

n 

.08 

.0011 

5 

.0006 

111 

.10 

.0017 

6 

7 

.0083 

17 
10 

.12 

+   .0024 

—   .0099 

.14 

.0033 

9 

.0110 

17 

.10 

.0043 

■    10 

.0132 

16 

.18 

.0054 

11 

.0148 

16 

.20 

.0067 

13 

14 

.0103 

15 

10 

.22 

+   .0081 

-   .0179 

.24 

.0096 

15 

.0194 

15 

.20 

.0113 

17 

.0209 

15 

.28 

.:jo 

.0131 

.0150 

18 
19 

21 

.0224 
.0239 

15 
15 

14 

.32 

+   .0171 

-   .0253 

.34 
.36 

.0193 
.0216 

22 
23 

.0207 
.0281 

14 
14 

.38 

.0241 

25 

.0294 

13 

.40 

.0267 

26 

27 

.0307 

13 

12 

.42 

+   .0294 

-   .0319 

.44 

.0323 

29 

.0331 

12 

.40 

.0353 

30 

.0343 

12 

.48 

.0384 

31 

.0354 

11 

.50 

.0417 

33 
34 

.0365 

11 
10 

.52 

+   .0451 

-   .0375 

9 
9 

.54 
.50 

.0486 
.0523 

35 

37 

.0384 
.0393 

.58 

.0501 

38 

.0402 

9 

0.00 

+  0.0000 

+39 

-0.0410 

—  8 

BillLIOGRAPIlY. 

List  of  "the  Principal  Papkbs,  Memoires,  etc..  upon  the  Subjects  of  Intekpolation 

AND  Mechanical  Quadrature. 


Astrand  (J.  J.).     Vierteljalu-sschrift  der  Astrouomischeu  Gesellschaft,  Vol.  X  (1875), 

p.  279. 
Baillaud  (B.).     Annales  de  I'Observatoire  de  Toulouse,  Vol.  II,  p.  B.l. 
Bienayme  (Jules).     Liouville,  Journal  de  Mathematiques,  Vol.  XVIII  (1853),  p.  299. 
Boole  (George).     A  Treatise  on  the  Calculus  of  Finite  Differences,  Cliai)ters  II  and  III. 
Brassinne  (E.).     Liouville,  Journal  de  Mathematiques,  Vol.  XI  (1846),  p.  177. 
Briinnow  (F.).     Lehrbucli  der  Spharischen  Astronomie,  p.  18. 

Cauchy  (Augustin).     (a)  Liouville,  Journal  de  Mathematiques,  Vol.  II  (1837),  p.  193. 
(Jj)     Comptes  Eeudus,  Vol.  XI  (1840),  p.  775. 
(r)     Ibid.,  Vol.  XIX  (1844),  p.  1183. 
{d)     Connaissance  des  Temps,  1852  (Additions),  p.  129. 

Chauvenet  (Wm.).     Spherical  and  Practical  Astronomy,  Vol.  I,  p.  79. 

Davis  (C.  H.).     The  ISIathematical  Monthly  (Cambridge,  Mass.),  Vol.  II  (1860),  p.  276. 

Doolittle  (C  L.).     A  Treatise  on  Practical  Astronomy,  p.  70. 

Encke  (J.  F.).      («)  Berliner  Astronomisches  Jahrbuch,  1830,  p.  265. 

(6)  Ibid.,  1837,  p.  251. 

(c)  Ibid.,  1852,  p.  330. 

{d)     Ibid.,  1862,  p.  313. 

Ferrel  (William).      The  Mathematical  Monthly  (Cambridge,  Mass.),  Vol.  Ill  (1861), 

p.  377. 
Gauss  (Carl  F.).     Werke,  Vol.  Ill,  p.  265. 
Grunert  (J.  A.),     (a)     Archiv  der  Mathematik  und  I'hysik,  Vol.  XIV  (1850),  p.  225. 

{h)     Ibid.,  Vol.  XX  (1853),  p.  361. 
Hansen  (P.  A.),     (a)  Abhandlungen  der  Koniglich  Sachsischen  Gesellschaft  der  Wis- 
senschaften  (Leipzig),  Vol.  XI  (1865),  p.  505. 
{b)     Tables  de  la  Lune,  p.  68. 

Herr  und  Tinter.     Lehrbuch  der  Spharischen  Astronomie,  Chapter  II. 

Jacobi  (C.  G.  J.),     (a)  Crelle's  Journal,  Vol.  I  (1826),  p.  301. 

(h)     Ibid.,  VoL  XXX  (1846),  p.  127. 
Klinkerfues  (W.).     Theoretisehe  Astronomie  (2d  edition,  1899),  pp.  67  and  490. 

233 


234:  nii5LiO(;RAiMiY. 

Kliigel's  Matliematisches  Worterbuch.      See   article  "Eiiisclialten,'"  wliicli  inchuk's   a 
brief  history  of  the  subject. 

Lagrange  (J.  L.).     («)  CEuvres,  Vol.  V,  p.  ()G3. 
{/,)     Ibid,  Vol.  YII,  p.  535. 

Laplace  (P.  S.).     Mecauique  Celeste,  Vol.  IV,  pp.  204-207. 

Le  Verrier  (U.  J.).      Annales  de    TObservatoire  de  Paris  (Memoires),  Vol.1,  jip.  121, 
129,  151,  154. 

Loomis  (Elias).     Practical  Astronomy,  p.  202. 

Maurice  (Fred.).     Connaissance  des  Temps,  1847  (Additions),  p.  181. 

Merrifleld  (C.  W.).     British  Association  Eeport,  Vol.  L  (1880),  p.  321. 

Newcomb  (Simon).     Logarithmic  and  Other  Mathematical  Tables,  p.  56. 

Newton  (Isaac).     Prinoipia,  Book  III,  Lemma  V. 

Olivier  (Louis).     Crelle's  Journal,  Vol.  II  (1827),  p.  252. 

Oppolzer  (T.  R.).     Lehrbuch  zur  Bahnbestimmuug,  Vol.  II,  pp.  1  and  596. 

Radau  (Rodolphe).       (a)     Liouville,  Journal  de  Mathematiques,  3d  Series,  Vol.  VI 
(1880),  p.  283. 
(i).     Bulletin  Astronomiqiie,  Vol.  VIII  (1891),  pp.  273,  325,  376,  425. 
Rees's  Cyclopedia.     See  article  "Interpolation." 

Sawitsch  (A.).     Abriss  der  Practischen  Astronomie,  Vol.11,  p.  416;    or  see    the  one 

volume  edition,  p.  818. 
Tisserand  (F.).     (a)  Comptes  Eendus,  Vol.  LXVIII  (1869),  p.  1101. 
{b)     Ibid.,  Vol.  LXX  (1870),  p.  678. 
^  (c)      Traite  de  Mecanique  Celeste,  Vol.  IV,  Chapters  X  and  XI. 

Valentiner  (W.).     Handworterbuch  der  Astronomie,  Vol.  II,  pp.  41  and  618. 
Watson  (James  C).     Theoretical  Astronomy,  pp.  112,  335,  435. 
Weddle  (Thomas).      Cambridge  and  Dublin    Mathematical  Journal,  Vol.  IX  (1854), 
p.  79. 


x^ 


SARNIE'S 
U08)<ERY 

72!  l. 
SAN  «IE&t 

CiM. 


UC  SOUTHERN  Rf GIONAI  I IBRARV  FACILITY 

D    001  113  399   8 


NOV  U 


1978 


